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== 7.6#h(0.6em)Quadratic Equations

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Solve: #math.equation(block: false, alt: "5 y minus 3 equals 0")[$5 y − 3 = 0$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "y equals the fraction 3 over 5")[$y = frac(3, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "10 a equals 0")[$10 a = 0$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "a equals 0")[$a = 0$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Combine like terms: #math.equation(block: false, alt: "12 x squared minus 6 x plus 4 x")[$12 x^(2) − 6 x + 4 x$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "12 x squared minus 2 x")[$12 x^(2) − 2 x$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Factor #math.equation(block: false, alt: "n cubed minus 9 n squared minus 22 n")[$n^(3) − 9 n^(2) − 22 n$] completely. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "n open parenthesis n minus 11 close parenthesis open parenthesis n plus 2 close parenthesis")[$n ( n − 11 ) ( n + 2 )$]
]
]

We have already solved linear equations, equations of the form #math.equation(block: false, alt: "a x plus b y equals c")[$a x + b y = c$]. In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. Listed below are some examples of quadratic equations:

#math.equation(block: true, alt: "x squared plus 5 x plus 6 equals 0, 3 y squared plus 4 y equals 10, 64 u squared minus 81 equals 0, n open parenthesis n plus 1 close parenthesis equals 42")[$x^(2) + 5 x + 6 = 0 &  &  & 3 y^(2) + 4 y = 10 &  &  & 64 u^(2) − 81 = 0 &  &  & n ( n + 1 ) = 42$]The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get #math.equation(block: false, alt: "n squared plus n")[$n^(2) + n$].

The general form of a quadratic equation is #math.equation(block: false, alt: "a x squared plus b x plus c equals 0 , with a not equal to 0")[$a x^(2) + b x + c = 0 , "with" #h(0.2em) a ≠ 0$].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Quadratic Equation]

An equation of the form #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$] is called a quadratic equation.

#math.equation(block: true, alt: "a , b , and c are real numbers and a not equal to 0")[$a , b , #h(0.2em) "and" #h(0.2em) c #h(0.2em) "are real numbers and" #h(0.2em) a ≠ 0$]
]

To solve #strong[quadratic equations] we need methods different than the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.


=== Solve Quadratic Equations Using the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The #strong[Zero Product Property] says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Zero Product Property]

If #math.equation(block: false, alt: "a times b equals 0")[$a · b = 0$], then either #math.equation(block: false, alt: "a equals 0")[$a = 0$] or #math.equation(block: false, alt: "b equals 0")[$b = 0$] or both.
]

We will now use the #strong[Zero Product Property], to solve a quadratic equation.

#examplebox("Example 1")[How to Use the Zero Product Property to Solve a Quadratic Equation][
Solve: #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis open parenthesis x minus 4 close parenthesis equals 0")[$( x + 1 ) ( x − 4 ) = 0$].

#solutionbox[
#figure(figph[This table gives the steps for solving (x + 1)(x – 4) = 0. The first step is to set each factor equal to 0. Since it is a product equal to 0, at least one factor must equal 0. x + 1 = 0 or x – 4 = 0.], alt: "This table gives the steps for solving (x + 1)(x – 4) = 0. The first step is to set each factor equal to 0. Since it is a product equal to 0, at least one factor must equal 0. x + 1 = 0 or x – 4 = 0.", caption: none)

#figure(figph[The next step is to solve each linear equation. This gives two solutions, x = −1 or x = 4.], alt: "The next step is to solve each linear equation. This gives two solutions, x = −1 or x = 4.", caption: none)

#figure(figph[The last step is to check both answers by substituting the values for x into the original equation.], alt: "The last step is to check both answers by substituting the values for x into the original equation.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 5 close parenthesis equals 0")[$( x − 3 ) ( x + 5 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "x equals 3 , x equals −5")[$x = 3 , x = −5$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis y minus 6 close parenthesis open parenthesis y plus 9 close parenthesis equals 0")[$( y − 6 ) ( y + 9 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "y equals 6 , y equals −9")[$y = 6 , y = −9$]
]
]

We usually will do a little more work than we did in this last example to solve the linear equations that result from using the Zero Product Property.

#examplebox("Example 2")[][
Solve: #math.equation(block: false, alt: "open parenthesis 5 n minus 2 close parenthesis open parenthesis 6 n minus 1 close parenthesis equals 0")[$( 5 n − 2 ) ( 6 n − 1 ) = 0$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "open parenthesis 5 n minus 2 close parenthesis open parenthesis 6 n minus 1 close parenthesis equals 0")[$( 5 n − 2 ) ( 6 n − 1 ) = 0$]], [],
  [Use the Zero Product Property to set #linebreak() each factor to 0.], [#math.equation(block: false, alt: "5 n minus 2 equals 0")[$5 n − 2 = 0$]], [#math.equation(block: false, alt: "6 n minus 1 equals 0")[$6 n − 1 = 0$]],
  [Solve the equations.], [#math.equation(block: false, alt: "n equals the fraction 2 over 5")[$n = frac(2, 5)$]], [#math.equation(block: false, alt: "n equals the fraction 1 over 6")[$n = frac(1, 6)$]],
  [Check your answers.], [], [],
  [#figure(figph[Verification of solutions for the quadratic equation (5n - 2)(6n - 1) = 0. Both n = 2/5 and n = 1/6 are substituted into the equation, confirming that both values result in 0 = 0.], alt: "Verification of solutions for the quadratic equation (5n - 2)(6n - 1) = 0. Both n = 2/5 and n = 1/6 are substituted into the equation, confirming that both values result in 0 = 0.", caption: none)], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis 3 m minus 2 close parenthesis open parenthesis 2 m plus 1 close parenthesis equals 0")[$( 3 m − 2 ) ( 2 m + 1 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "m equals the fraction 2 over 3 , m equals minus the fraction 1 over 2")[$m = frac(2, 3) , m = − frac(1, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis 4 p plus 3 close parenthesis open parenthesis 4 p minus 3 close parenthesis equals 0")[$( 4 p + 3 ) ( 4 p − 3 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "p equals minus the fraction 3 over 4 , p equals the fraction 3 over 4")[$p = − frac(3, 4) , p = frac(3, 4)$]
]
]

Notice when we checked the solutions that each of them made just one factor equal to zero. But the product was zero for both solutions.

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "3 p open parenthesis 10 p plus 7 close parenthesis equals 0")[$3 p ( 10 p + 7 ) = 0$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "3 p open parenthesis 10 p plus 7 close parenthesis equals 0")[$#h(1.2em) 3 p ( 10 p + 7 ) = 0$]], [],
  [Use the Zero Product Property to set #linebreak() each factor to 0.], [#math.equation(block: false, alt: "3 p equals 0")[$3 p = 0$]], [#math.equation(block: false, alt: "10 p plus 7 equals 0")[$10 p + 7 = 0 #h(1.6em)$]],
  [Solve the equations.], [#math.equation(block: false, alt: "p equals 0")[$p = 0$]], [#math.equation(block: false, alt: "10 p equals −7")[$10 p = −7 #h(0.9em)$]],
  [], [], [#math.equation(block: false, alt: "p equals minus the fraction 7 over 10")[$p = − frac(7, 10)$]],
  [Check your answers.], [], [],
  [#figure(figph[Solutions p=0 and p=-7/10 are verified for the equation 3p(10p + 7)=0. Both substitutions lead to 0=0, confirming their correctness.], alt: "Solutions p=0 and p=-7/10 are verified for the equation 3p(10p + 7)=0. Both substitutions lead to 0=0, confirming their correctness.", caption: none)], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 u open parenthesis 5 u minus 1 close parenthesis equals 0")[$2 u ( 5 u − 1 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "u equals 0 , u equals the fraction 1 over 5")[$u = 0 , u = frac(1, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "w open parenthesis 2 w plus 3 close parenthesis equals 0")[$w ( 2 w + 3 ) = 0$].

#solutionbox[
#math.equation(block: true, alt: "w equals 0 , w equals minus the fraction 3 over 2")[$w = 0 , w = − frac(3, 2)$]
]
]

It may appear that there is only one factor in the next example. Remember, however, that #math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis squared")[$attach(( y − 8 ), t: 2)$] means #math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis open parenthesis y minus 8 close parenthesis")[$( y − 8 ) ( y − 8 )$].

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis squared equals 0")[$attach(( y − 8 ), t: 2) = 0$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis squared equals 0")[$attach(( y − 8 ), t: 2) = 0$]], [],
  [Rewrite the left side as a product.], [#math.equation(block: false, alt: "open parenthesis y minus 8 close parenthesis open parenthesis y minus 8 close parenthesis equals 0")[$( y − 8 ) ( y − 8 ) = 0$]], [],
  [Use the Zero Product Property and #linebreak() set each factor to 0.], [#math.equation(block: false, alt: "y minus 8 equals 0")[$y − 8 = 0$]], [#math.equation(block: false, alt: "y minus 8 equals 0")[$y − 8 = 0$]],
  [Solve the equations.], [#math.equation(block: false, alt: "y equals 8")[$y = 8$]], [#math.equation(block: false, alt: "y equals 8")[$y = 8$]],
  [When a solution repeats, we call it #linebreak() a double root.], [], [],
  [Check your answer.], [], [],
  [#figure(figph[A step-by-step verification of the equation (y-8)^2=0 by substituting y=8, leading to the true statement 0=0.], alt: "A step-by-step verification of the equation (y-8)^2=0 by substituting y=8, leading to the true statement 0=0.", caption: none)], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis x plus 1 close parenthesis squared equals 0")[$attach(( x + 1 ), t: 2) = 0$].

#solutionbox[
#math.equation(block: true, alt: "x equals −1")[$x = −1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis v minus 2 close parenthesis squared equals 0")[$attach(( v − 2 ), t: 2) = 0$].

#solutionbox[
#math.equation(block: true, alt: "v equals 2")[$v = 2$]
]
]


=== Solve Quadratic Equations by Factoring

Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we must be sure to start with the quadratic equation in standard form, #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$]. Then we can factor the expression on the left.

#examplebox("Example 5")[How to Solve a Quadratic Equation by Factoring][
Solve: #math.equation(block: false, alt: "x squared plus 2 x minus 8 equals 0")[$x^(2) + 2 x − 8 = 0$].

#solutionbox[
#figure(figph[This table gives the steps for solving the equation x squared + 2 x – 8 = 0. The first step is writing the equation in standard quadratic form, which it is.], alt: "This table gives the steps for solving the equation x squared + 2 x – 8 = 0. The first step is writing the equation in standard quadratic form, which it is.", caption: none)

#figure(figph[The second step is factoring the quadratic expression x squared + 2 x – 8. The factors are (x + 4), (x – 2).], alt: "The second step is factoring the quadratic expression x squared + 2 x – 8. The factors are (x + 4), (x – 2).", caption: none)

#figure(figph[The next step is using the zero product property and set each factor equal to 0, x + 4 = 0 and x – 2 = 0.], alt: "The next step is using the zero product property and set each factor equal to 0, x + 4 = 0 and x – 2 = 0.", caption: none)

#figure(figph[The next step is solving both linear equations, x = −4 or x = 2.], alt: "The next step is solving both linear equations, x = −4 or x = 2.", caption: none)

#figure(figph[The last step is checking both solutions by substituting them into the original equation.], alt: "The last step is checking both solutions by substituting them into the original equation.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "x squared minus x minus 12 equals 0")[$x^(2) − x − 12 = 0$].

#solutionbox[
#math.equation(block: true, alt: "x equals 4 , x equals −3")[$x = 4 , x = −3$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "b squared plus 9 b plus 14 equals 0")[$b^(2) + 9 b + 14 = 0$].

#solutionbox[
#math.equation(block: true, alt: "b equals −2 , b equals −7")[$b = −2 , b = −7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve a quadratic equation by factoring.]

+ Write the quadratic equation in standard form, #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$].
+ Factor the quadratic expression.
+ Use the Zero Product Property.
+ Solve the linear equations.
+ Check.
]

Before we factor, we must make sure the quadratic equation is in standard form.

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "2 y squared equals 13 y plus 45")[$2 y^(2) = 13 y + 45$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 y squared equals 13 y plus 45")[$#h(4.65em) 2 y^(2) = 13 y + 45$]], []),
  [Write the quadratic equation in standard form.], [#math.equation(block: false, alt: "2 y squared minus 13 y minus 45 equals 0")[$2 y^(2) − 13 y − 45 = 0$]], [],
  [Factor the quadratic expression.], [#math.equation(block: false, alt: "open parenthesis 2 y plus 5 close parenthesis open parenthesis y minus 9 close parenthesis equals 0")[$#h(0.7em) ( 2 y + 5 ) ( y − 9 ) = 0$]], [],
  [Use the Zero Product Property #linebreak() to set each factor to 0.], [#math.equation(block: false, alt: "2 y plus 5 equals 0")[$#h(3.46em) 2 y + 5 = 0$]], [#math.equation(block: false, alt: "y minus 9 equals 0")[$y − 9 = 0$]],
  [Solve each equation.], [#math.equation(block: false, alt: "y equals minus the fraction 5 over 2")[$#h(5.68em) y = − frac(5, 2)$]], [#math.equation(block: false, alt: "y equals 9")[$y = 9$]],
  [Check your answers.], [], [],
  [#figure(figph[Step-by-step verification of two solutions, y = -5/2 and y = 9, for the quadratic equation 2y^2 = 13y + 45, confirming both satisfy the equation.], alt: "Step-by-step verification of two solutions, y = -5/2 and y = 9, for the quadratic equation 2y^2 = 13y + 45, confirming both satisfy the equation.", caption: none)], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 c squared equals 10 c minus 8")[$3 c^(2) = 10 c − 8$].

#solutionbox[
#math.equation(block: true, alt: "c equals 2 , c equals the fraction 4 over 3")[$c = 2 , c = frac(4, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 d squared minus 5 d equals 3")[$2 d^(2) − 5 d = 3$].

#solutionbox[
#math.equation(block: true, alt: "d equals 3 , d equals minus the fraction 1 over 2")[$d = 3 , d = − frac(1, 2)$]
]
]

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "5 x squared minus 13 x equals 7 x")[$5 x^(2) − 13 x = 7 x$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "5 x squared minus 13 x equals 7 x")[$5 x^(2) − 13 x = 7 x$]], []),
  [Write the quadratic equation in standard form.], [#math.equation(block: false, alt: "5 x squared minus 20 x equals 0")[$5 x^(2) − 20 x = 0 #h(0.5em)$]], [],
  [Factor the left side of the equation.], [#math.equation(block: false, alt: "5 x open parenthesis x minus 4 close parenthesis equals 0")[$5 x ( x − 4 ) = 0 #h(0.5em)$]], [],
  [Use the Zero Product Property #linebreak() to set each factor to 0.], [#math.equation(block: false, alt: "5 x equals 0")[$5 x = 0 #h(0.5em)$]], [#math.equation(block: false, alt: "x minus 4 equals 0")[$x − 4 = 0$]],
  [Solve each equation.], [#math.equation(block: false, alt: "x equals 0")[$x = 0 #h(0.5em)$]], [#math.equation(block: false, alt: "x equals 4")[$x = 4$]],
  [Check your answers.], [], [],
  [#figure(figph[The image shows the verification of potential solutions for the equation 5x^2 - 13x = 7x. It demonstrates substituting x=0 and x=4 into the equation, showing that both values satisfy the equation.], alt: "The image shows the verification of potential solutions for the equation 5x^2 - 13x = 7x. It demonstrates substituting x=0 and x=4 into the equation, showing that both values satisfy the equation.", caption: none)], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "6 a squared plus 9 a equals 3 a")[$6 a^(2) + 9 a = 3 a$].

#solutionbox[
#math.equation(block: true, alt: "a equals 0 , a equals −1")[$a = 0 , a = −1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "45 b squared minus 2 b equals −17 b")[$45 b^(2) − 2 b = −17 b$].

#solutionbox[
#math.equation(block: true, alt: "b equals 0 , b equals minus the fraction 1 over 3")[$b = 0 , b = − frac(1, 3)$]
]
]

Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "144 q squared equals 25")[$144 q^(2) = 25$].

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background, reading 144q squared equals 25.], alt: "A mathematical equation is displayed on a white background, reading 144q squared equals 25.", caption: none)], []),
  [Write the quadratic equation in standard form.], [#figure(figph[A mathematical equation, 144q^2 - 25 = 0, is displayed on a white background.], alt: "A mathematical equation, 144q^2 - 25 = 0, is displayed on a white background.", caption: none)], [],
  [Factor. It is a difference of squares.], [#figure(figph[A mathematical equation is displayed against a white background, reading (12q - 5)(12q + 5) = 0. The equation is presented in black text.], alt: "A mathematical equation is displayed against a white background, reading (12q - 5)(12q + 5) = 0. The equation is presented in black text.", caption: none)], [],
  [Use the Zero Product Property to set each factor to 0.], [#math.equation(block: false, alt: "12 q minus 5 equals 0")[$12 q − 5 = 0$]], [#math.equation(block: false, alt: "12 q plus 5 equals 0")[$12 q + 5 = 0$]],
  [Solve each equation.], [#math.equation(block: false, alt: "12 q, equals, 5; q, equals, the fraction 5 over 12")[$12 q & = & 5 \ q & = & frac(5, 12)$]], [#math.equation(block: false, alt: "12 q, equals, –5; q, equals, minus the fraction 5 over 12")[$12 q & = & –5 \ q & = & − frac(5, 12)$]],
  [Check your answers.], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "25 p squared equals 49")[$25 p^(2) = 49$].

#solutionbox[
#math.equation(block: true, alt: "p equals the fraction 7 over 5 , p equals minus the fraction 7 over 5")[$p = frac(7, 5) , "p" = − frac(7, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "36 x squared equals 121")[$36 x^(2) = 121$].

#solutionbox[
#math.equation(block: true, alt: "x equals the fraction 11 over 6 , x equals minus the fraction 11 over 6")[$x = frac(11, 6) , x = − frac(11, 6)$]
]
]

The left side in the next example is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.

#examplebox("Example 9")[][
Solve: #math.equation(block: false, alt: "open parenthesis 3 x minus 8 close parenthesis open parenthesis x minus 1 close parenthesis equals 3 x")[$( 3 x − 8 ) ( x − 1 ) = 3 x$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 x minus 8 close parenthesis open parenthesis x minus 1 close parenthesis, equals, 3 x")[$( 3 x − 8 ) ( x − 1 ) & = & 3 x$]]),
  [Multiply the binomials.], [#math.equation(block: false, alt: "3 x squared minus 11 x plus 8, equals, 3 x")[$#h(0.5em) 3 x^(2) − 11 x + 8 & = & 3 x$]],
  [Write the quadratic equation in standard form.], [#math.equation(block: false, alt: "3 x squared minus 14 x plus 8, equals, 0")[$#h(0.5em) 3 x^(2) − 14 x + 8 & = & 0$]],
  [Factor the trinomial.], [#math.equation(block: false, alt: "open parenthesis 3 x minus 2 close parenthesis open parenthesis x minus 4 close parenthesis, equals, 0")[$#h(0.2em) ( 3 x − 2 ) ( x − 4 ) & = & 0$]],
  [Use the Zero Product Property to set each factor to 0. #linebreak() Solve each equation.], [#math.equation(block: false, alt: "3 x minus 2, equals, 0, x minus 4, equals, 0; 3 x, equals, 2, x, equals, 4; x, equals, the fraction 2 over 3")[$#h(3.6em) 3 x − 2 & = & 0 &  &  & x − 4 & = & 0 \ 3 x & = & 2 &  &  & x & = & 4 \ x & = & frac(2, 3)$]],
  [Check your answers.], [The check is left to you!],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis 2 m plus 1 close parenthesis open parenthesis m plus 3 close parenthesis equals 12 m")[$( 2 m + 1 ) ( m + 3 ) = 12 m$].

#solutionbox[
#math.equation(block: true, alt: "m equals 1 , m equals the fraction 3 over 2")[$m = 1 , m = frac(3, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "open parenthesis k plus 1 close parenthesis open parenthesis k minus 1 close parenthesis equals 8")[$( k + 1 ) ( k − 1 ) = 8$].

#solutionbox[
#math.equation(block: true, alt: "k equals 3 , k equals −3")[$k = 3 , k = −3$]
]
]

The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree more than two by using the Zero Product Property, just like we solved quadratic equations.

#examplebox("Example 10")[][
Solve: #math.equation(block: false, alt: "9 m cubed plus 100 m equals 60 m squared")[$9 m^(3) + 100 m = 60 m^(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation displayed on a white background: 9m^3 + 100m = 60m^2.], alt: "A mathematical equation displayed on a white background: 9m^3 + 100m = 60m^2.", caption: none)]),
  [Bring all the terms to one side so that the other side is zero.], [#figure(figph[A cubic equation is displayed: 9m^3 - 60m^2 + 100m = 0, featuring terms with m raised to the power of 3, 2, and 1, set equal to zero.], alt: "A cubic equation is displayed: 9m^3 - 60m^2 + 100m = 0, featuring terms with m raised to the power of 3, 2, and 1, set equal to zero.", caption: none)],
  [Factor the greatest common factor first.], [#figure(figph[A mathematical equation is displayed, showing m multiplied by the quadratic expression (9m^2 - 60m + 100), all set equal to zero.], alt: "A mathematical equation is displayed, showing m multiplied by the quadratic expression (9m^2 - 60m + 100), all set equal to zero.", caption: none)],
  [Factor the trinomial.], [#figure(figph[A mathematical equation is displayed on a white background: m(3m - 10)(3m - 10) = 0. The equation shows a variable 'm' multiplied by two identical binomial factors (3m - 10), all set equal to zero.], alt: "A mathematical equation is displayed on a white background: m(3m - 10)(3m - 10) = 0. The equation shows a variable 'm' multiplied by two identical binomial factors (3m - 10), all set equal to zero.", caption: none)],
  [Use the Zero Product Property to set each factor to 0.], [#figure(figph[Three mathematical equations are displayed horizontally on a white background: 'm = 0', '3m - 10 = 0', and '3m - 10 = 0'.], alt: "Three mathematical equations are displayed horizontally on a white background: 'm = 0', '3m - 10 = 0', and '3m - 10 = 0'.", caption: none)],
  [Solve each equation.], [#figure(figph[Three mathematical expressions are shown: m = 0, m = 10/3, and m = 10/3. The letter 'm' is equal to zero in the first expression and to the fraction 10 over 3 in the other two.], alt: "Three mathematical expressions are shown: m = 0, m = 10/3, and m = 10/3. The letter 'm' is equal to zero in the first expression and to the fraction 10 over 3 in the other two.", caption: none)],
  [Check your answers.], [The check is left to you.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "8 x cubed equals 24 x squared minus 18 x")[$8 x^(3) = 24 x^(2) − 18 x$].

#solutionbox[
#math.equation(block: true, alt: "x equals 0 , x equals the fraction 3 over 2")[$x = 0 , x = frac(3, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "16 y squared equals 32 y cubed plus 2 y")[$16 y^(2) = 32 y^(3) + 2 y$].

#solutionbox[
#math.equation(block: true, alt: "y equals 0 , y equals the fraction 1 over 4")[$y = 0 , y = frac(1, 4)$]
]
]

When we factor the quadratic equation in the next example we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

#examplebox("Example 11")[][
Solve: #math.equation(block: false, alt: "4 x squared equals 16 x plus 84")[$4 x^(2) = 16 x + 84$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 x squared, equals, 16 x plus 84")[$#h(5em) 4 x^(2) & = & 16 x + 84$]]),
  [Write the quadratic equation in standard form.], [#math.equation(block: false, alt: "4 x squared minus 16 x minus 84, equals, 0")[$#h(0.3em) 4 x^(2) − 16 x − 84 & = & 0$]],
  [Factor the greatest common factor first.], [#math.equation(block: false, alt: "4 open parenthesis x squared minus 4 x minus 21 close parenthesis, equals, 0")[$#h(0.3em) 4 ( x^(2) − 4 x − 21 ) & = & 0$]],
  [Factor the trinomial.], [#math.equation(block: false, alt: "4 open parenthesis x minus 7 close parenthesis open parenthesis x plus 3 close parenthesis, equals, 0")[$#h(0.5em) 4 ( x − 7 ) ( x + 3 ) & = & 0$]],
  [Use the Zero Product Property to set each factor to 0. #linebreak() Solve each equation.], [#math.equation(block: false, alt: "4, not equal to, 0, x minus 7, equals, 0, x plus 3, equals, 0; 4, not equal to, 0, x, equals, 7, x, equals, −3")[$#h(0.4em) 4 & ≠ & 0 &  &  & x − 7 & = & 0 &  &  & x + 3 & = & 0 \ 4 & ≠ & 0 &  &  & x & = & 7 &  &  & x & = & −3$]],
  [Check your answers.], [The check is left to you.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "18 a squared minus 30 equals −33 a")[$18 a^(2) − 30 = −33 a$].

#solutionbox[
#math.equation(block: true, alt: "a equals minus the fraction 5 over 2 , a equals the fraction 2 over 3")[$a = − frac(5, 2) , a = frac(2, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "123 b equals −6 minus 60 b squared")[$123 b = −6 − 60 b^(2)$].

#solutionbox[
#math.equation(block: true, alt: "b equals −2 , b equals minus the fraction 1 over 20")[$b = −2 , b = − frac(1, 20)$]
]
]


=== Solve Applications Modeled by Quadratic Equations

The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use a problem-solving strategy to solve word problems.]

+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.
]

We will start with a number problem to get practice translating words into a quadratic equation.

#examplebox("Example 12")[][
The product of two consecutive integers is 132. Find the integers.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem.], [],
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for two consecutive integers.],
  [#strong[Step 3. Name] what we are looking for.], [#math.equation(block: false, alt: "Let n equals the first integer; n plus 1 equals the next consecutive integer")[$"Let" #h(0.2em) n = "the first integer" \ n + 1 = #h(0.2em) "the next consecutive integer"$]],
  [#strong[Step 4. Translate] into an equation. Restate the problem in a sentence.], [The product of the two consecutive integers is 132.],
  [], [The first integer times the next integer is 132.],
  [Translate to an equation.], [#math.equation(block: false, alt: "n open parenthesis n plus 1 close parenthesis, equals, 132")[$#h(3.2em) n ( n + 1 ) & = & 132$]],
  [#strong[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "n squared plus n, equals, 132")[$#h(4em) n^(2) + n & = & 132$]],
  [Bring all the terms to one side.], [#math.equation(block: false, alt: "n squared plus n minus 132, equals, 0")[$#h(1.4em) n^(2) + n − 132 & = & 0$]],
  [Factor the trinomial.], [#math.equation(block: false, alt: "open parenthesis n minus 11 close parenthesis open parenthesis n plus 12 close parenthesis, equals, 0")[$( n − 11 ) ( n + 12 ) & = & 0$]],
  [Use the zero product property. #linebreak() Solve the equations.], [#math.equation(block: false, alt: "n minus 11, equals, 0, n plus 12, equals, 0; n, equals, 11, n, equals, −12")[$#h(0.4em) #h(4em) n − 11 & = & 0 &  &  &  &  & n + 12 & = & 0 \ n & = & 11 &  &  &  &  & n & = & −12$]],
  [There are two values for #math.equation(block: false, alt: "n")[$n$] that are solutions to this problem. So there are two sets of consecutive integers that will work.], [],
  [#math.equation(block: false, alt: "If the first integer is n equals 11, If the first integer is n equals −12; then the next integer is n plus 1, then the next integer is n plus 1; 11 plus 1, −12 plus 1; 12, −11")[$"If the first integer is" #h(0.2em) n = 11 &  &  &  &  & "If the first integer is" #h(0.2em) n = −12 \  \  \ "then the next integer is" #h(0.2em) n + 1 &  &  &  &  & "then the next integer is" #h(0.2em) n + 1 \  \  \ 11 + 1 &  &  &  &  & −12 + 1 \  \  \ 12 &  &  &  &  & −11$]], [],
  [#strong[Step 6. Check] the answer.], [],
  [The consecutive integers are #math.equation(block: false, alt: "11 , 12")[$11 , 12$] and #math.equation(block: false, alt: "−11 , −12")[$−11 , −12$]. The product #math.equation(block: false, alt: "11 times 12 equals 132")[$11 · 12 = 132$] and the product #math.equation(block: false, alt: "−11 open parenthesis −12 close parenthesis equals 132")[$−11 ( −12 ) = 132$]. Both pairs of consecutive integers are solutions.], [],
  [#strong[Step 7. Answer] the question. The consecutive integers are #math.equation(block: false, alt: "11 , 12")[$11 , 12$] and #math.equation(block: false, alt: "−11 , −12")[$−11 , −12$].], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The product of two consecutive integers is #math.equation(block: false, alt: "240")[$240$]. Find the integers.

#solutionbox[
#math.equation(block: true, alt: "−15 , −16 and 15 , 16")[$−15 , −16 #h(0.2em) "and" #h(0.2em) 15 , 16$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The product of two consecutive integers is #math.equation(block: false, alt: "420")[$420$]. Find the integers.

#solutionbox[
#math.equation(block: true, alt: "−21 , −20 and 20 , 21")[$−21 , −20 #h(0.2em) "and" #h(0.2em) 20 , 21$]
]
]

Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give 132.

In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.

#examplebox("Example 13")[][
A rectangular garden has an area #math.equation(block: false, alt: "15")[$15$] square feet. The length of the garden is two feet more than the width. Find the length and width of the garden.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem. In problems involving geometric figures, a sketch can help you visualize the situation.], [#figure(figph[An overhead view of a rectangular park or garden with green grass, several trees, two brown rock formations, and an oval blue pond. The dimensions are labeled 'W' for width and 'W + 2' for length.], alt: "An overhead view of a rectangular park or garden with green grass, several trees, two brown rock formations, and an oval blue pond. The dimensions are labeled 'W' for width and 'W + 2' for length.", caption: none)], [],
  [#strong[Step 2. Identify] what you are looking for.], [We are looking for the length and width.], [],
  [#strong[Step 3. Name] what you are looking for. #linebreak() The length is two feet more than width.], [Let #emph[W] = the width of the garden. #linebreak() #emph[W] + 2 = the length of the garden], [],
  [#strong[Step 4. Translate] into an equation. #linebreak() Restate the important information in a sentence.], [#linebreak() The area of the rectangular garden is 15 square feet.], [],
  [Use the formula for the area of a rectangle.], [#math.equation(block: false, alt: "A equals L times W")[$#h(0.8em) A = L · W$]], [],
  [Substitute in the variables.], [#math.equation(block: false, alt: "15 equals open parenthesis W plus 2 close parenthesis W")[$#h(0.5em) 15 = ( W + 2 ) W$]], [],
  [#strong[Step 5. Solve] the equation. Distribute first.], [#math.equation(block: false, alt: "15 equals W squared plus 2 W")[$#h(0.5em) 15 = W^(2) + 2 W$]], [],
  [Get zero on one side.], [#math.equation(block: false, alt: "0 equals W squared plus 2 W minus 15")[$#h(1em) 0 = W^(2) + 2 W − 15$]], [],
  [Factor the trinomial.], [#math.equation(block: false, alt: "0 equals open parenthesis W plus 5 close parenthesis open parenthesis W minus 3 close parenthesis")[$#h(1em) 0 = ( W + 5 ) ( W − 3 )$]], [],
  [Use the Zero Product Property.], [#math.equation(block: false, alt: "0 equals W plus 5")[$#h(1em) 0 = W + 5$]], [#math.equation(block: false, alt: "0 equals W minus 3")[$0 = W − 3$]],
  [Solve each equation.], [#math.equation(block: false, alt: "−5 equals W")[$#h(0.4em) −5 = W$]], [#math.equation(block: false, alt: "3 equals W")[$3 = W$]],
  [Since #emph[W] is the width of the garden, #linebreak() it does not make sense for it to be #linebreak() negative. We eliminate that value for #emph[W].], [#math.equation(block: false, alt: "−5 equals W")[$#h(0.35em) cancel(−5 = W)$] #linebreak()  #linebreak() #math.equation(block: false, alt: "W equals 3")[$#h(1em) W = 3$]], [#math.equation(block: false, alt: "3 equals W")[$3 = W$] #linebreak()  #linebreak() Width is 3 feet.],
  [Find the value of the length.], [#math.equation(block: false, alt: "W plus 2 equals length")[$#h(1em) W + 2 = "length"$]], [],
  [], [#math.equation(block: false, alt: "3 plus 2")[$#h(1em) 3 + 2$]], [],
  [], [#math.equation(block: false, alt: "5")[$#h(1.6em) 5$]], [Length is 5 feet.],
  [#strong[Step 6. Check] the answer. #linebreak() Does the answer make sense?], [], [],
  [#figure(figph[An illustration of a rectangular park with trees and a pond, accompanied by mathematical calculations demonstrating how to find its area. The width is 'W' and length is 'W+2', with W set to 3, resulting in an area of 15.], alt: "An illustration of a rectangular park with trees and a pond, accompanied by mathematical calculations demonstrating how to find its area. The width is 'W' and length is 'W+2', with W set to 3, resulting in an area of 15.", caption: none)], [], [],
  [], [Yes, this makes sense.], [],
  [#strong[Step 7. Answer] the question.], [The width of the garden is 3 feet #linebreak() and the length is 5 feet.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

#solutionbox[
#math.equation(block: false, alt: "5")[$5$] feet and #math.equation(block: false, alt: "6")[$6$] feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

#solutionbox[
#math.equation(block: false, alt: "12")[$12$] feet and #math.equation(block: false, alt: "15")[$15$] feet
]
]

In an earlier chapter, we used the Pythagorean Theorem #math.equation(block: false, alt: "open parenthesis a squared plus b squared equals c squared close parenthesis")[$( a^(2) + b^(2) = c^(2) )$]. It gave the relation between the legs and the hypotenuse of a right triangle.

#figure(figph[This figure is a right triangle.], alt: "This figure is a right triangle.", caption: none)

We will use this formula in the next example.

#examplebox("Example 14")[][
Justine wants to put a deck in the corner of her backyard in the shape of a right triangle, as shown below. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the deck.

#figure(figph[This figure is a right triangle. The vertical leg is labeled “x – 7”. the horizontal leg, the base, is labeled “x”. The hypotenuse is labeled “17”.], alt: "This figure is a right triangle. The vertical leg is labeled “x – 7”. the horizontal leg, the base, is labeled “x”. The hypotenuse is labeled “17”.", caption: none)

#solutionbox[
#figure(table(
  columns: 4, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem.], [], [], [],
  [#strong[Step 2. Identify] what you are looking for. #linebreak()], [We are looking for the lengths of the sides #linebreak() of the deck.], [], [],
  [#strong[Step 3. Name] what you are looking for. #linebreak() One side is 7 less than the other.], [Let #emph[x] = length of a side of the deck #linebreak() #emph[x] − 7 = length of other side], [], [],
  [#strong[Step 4. Translate] into an equation. #linebreak() Since this is a right triangle we can use the #linebreak() Pythagorean Theorem.], [#math.equation(block: false, alt: "a squared plus b squared equals c squared")[$#h(5.3em) a^(2) + b^(2) = c^(2)$] #linebreak()], [], [],
  [Substitute in the variables.], [#math.equation(block: false, alt: "x squared plus open parenthesis x minus 7 close parenthesis squared equals 17 squared")[$#h(3.05em) x^(2) + attach(( x − 7 ), t: 2) = 17^(2)$]], [], [],
  [#strong[Step 5. Solve] the equation.], [#math.equation(block: false, alt: "x squared plus x squared minus 14 x plus 49 equals 289")[$#h(0.6em) x^(2) + x^(2) − 14 x + 49 = 289$]], [], [],
  [Simplify.], [#math.equation(block: false, alt: "2 x squared minus 14 x plus 49 equals 289")[$#h(2.15em) 2 x^(2) − 14 x + 49 = 289$]], [], [],
  [It is a quadratic equation, so get zero on one side.], [#math.equation(block: false, alt: "2 x squared minus 14 x minus 240 equals 0")[$#h(1.7em) 2 x^(2) − 14 x − 240 = 0$]], [], [],
  [Factor the greatest common factor.], [#math.equation(block: false, alt: "2 open parenthesis x squared minus 7 x minus 120 close parenthesis equals 0")[$#h(1.5em) 2 ( x^(2) − 7 x − 120 ) = 0$]], [], [],
  [Factor the trinomial.], [#math.equation(block: false, alt: "2 open parenthesis x minus 15 close parenthesis open parenthesis x plus 8 close parenthesis equals 0")[$#h(1.8em) 2 ( x − 15 ) ( x + 8 ) = 0$]], [], [],
  [Use the Zero Product Property.], [#math.equation(block: false, alt: "2 not equal to 0")[$2 ≠ 0$]], [#math.equation(block: false, alt: "x minus 15 equals 0")[$#h(1.6em) x − 15 = 0$]], [#math.equation(block: false, alt: "x plus 8 equals 0")[$x + 8 = 0$]],
  [Solve.], [#math.equation(block: false, alt: "2 not equal to 0")[$2 ≠ 0$]], [#math.equation(block: false, alt: "x equals 15")[$#h(3.75em) x = 15$]], [#math.equation(block: false, alt: "x equals −8")[$#h(1.7em) x = −8$]],
  [Since #emph[x] is a side of the triangle, #math.equation(block: false, alt: "x equals −8")[$x = "−8"$] does not #linebreak() make sense.], [#math.equation(block: false, alt: "2 not equal to 0")[$2 ≠ 0$] #linebreak()], [#math.equation(block: false, alt: "x equals 15")[$#h(3.75em) x = 15$] #linebreak()], [#math.equation(block: false, alt: "x equals −8")[$#h(1.7em) cancel(x = −8)$] #linebreak()],
  [Find the length of the other side.], [], [], [],
  [If the length of one side is], [#figure(figph[The mathematical equation 'x = 15' displayed on a plain white background.], alt: "The mathematical equation 'x = 15' displayed on a plain white background.", caption: none)], [], [],
  [then the length of the other side is], [#figure(figph[A white background with the mathematical expression 'X-7' written in black text in the center.], alt: "A white background with the mathematical expression 'X-7' written in black text in the center.", caption: none)], [], [],
  [], [], [#figure(figph[The image shows the numbers 15-7. The '1' and '5' are in a reddish-orange hue, and the '-' and '7' are in black, all presented on a plain white background.], alt: "The image shows the numbers 15-7. The '1' and '5' are in a reddish-orange hue, and the '-' and '7' are in black, all presented on a plain white background.", caption: none)], [],
  [], [8 is the length of the other side.], [], [],
  [#strong[Step 6. Check] the answer. #linebreak() Do these numbers make sense?], [], [], [],
  [#figure(figph[A diagram showing a right triangle with sides x, x-7, and 17. The solution reveals x=15, and verification confirms 15^2 + 8^2 = 17^2, demonstrating the Pythagorean theorem.], alt: "A diagram showing a right triangle with sides x, x-7, and 17. The solution reveals x=15, and verification confirms 15^2 + 8^2 = 17^2, demonstrating the Pythagorean theorem.", caption: none)], [], [], [],
  [#strong[Step 7. Answer] the question.], [The sides of the deck are 8, 15, and 17 feet.], [], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the sail.

#solutionbox[
#math.equation(block: false, alt: "5")[$5$] feet and #math.equation(block: false, alt: "12")[$12$] feet
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg.

#solutionbox[
#math.equation(block: false, alt: "24")[$24$] feet and #math.equation(block: false, alt: "25")[$25$] feet
]
]


=== Key Concepts

- #strong[Zero Product Property] If #math.equation(block: false, alt: "a times b equals 0")[$a · b = 0$], then either #math.equation(block: false, alt: "a equals 0")[$a = 0$] or #math.equation(block: false, alt: "b equals 0")[$b = 0$] or both.
- #strong[Solve a quadratic equation by factoring] To solve a quadratic equation by factoring:
+ Write the quadratic equation in standard form, #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$].
+ Factor the quadratic expression.
+ Use the Zero Product Property.
+ Solve the linear equations.
+ Check.
- #strong[Use a problem solving strategy to solve word problems]
+ #strong[Read] the problem. Make sure all the words and ideas are understood.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for. Choose a variable to represent that quantity.
+ #strong[Translate] into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
+ #strong[Solve] the equation using good algebra techniques.
+ #strong[Check] the answer in the problem and make sure it makes sense.
+ #strong[Answer] the question with a complete sentence.


=== Section Exercises


==== Practice Makes Perfect

#strong[Use the Zero Product Property]

In the following exercises, solve.

#math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 7 close parenthesis equals 0")[$( x − 3 ) ( x + 7 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "x equals 3 , x equals −7")[$x = 3 , x = −7$]
]

#math.equation(block: true, alt: "open parenthesis y minus 11 close parenthesis open parenthesis y plus 1 close parenthesis equals 0")[$( y − 11 ) ( y + 1 ) = 0$]

#math.equation(block: true, alt: "open parenthesis 3 a minus 10 close parenthesis open parenthesis 2 a minus 7 close parenthesis equals 0")[$( 3 a − 10 ) ( 2 a − 7 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "a equals 10 / 3 , a equals 7 / 2")[$a = 10 "/" 3 , a = 7 "/" 2$]
]

#math.equation(block: true, alt: "open parenthesis 5 b plus 1 close parenthesis open parenthesis 6 b plus 1 close parenthesis equals 0")[$( 5 b + 1 ) ( 6 b + 1 ) = 0$]

#math.equation(block: true, alt: "6 m open parenthesis 12 m minus 5 close parenthesis equals 0")[$6 m ( 12 m − 5 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "m equals 0 , m equals 5 / 12")[$m = 0 , m = 5 "/" 12$]
]

#math.equation(block: true, alt: "2 x open parenthesis 6 x minus 3 close parenthesis equals 0")[$2 x ( 6 x − 3 ) = 0$]

#math.equation(block: true, alt: "open parenthesis y minus 3 close parenthesis squared equals 0")[$attach(( y − 3 ), t: 2) = 0$]

#solutionbox[
#math.equation(block: true, alt: "y equals 3")[$y = 3$]
]

#math.equation(block: true, alt: "open parenthesis b plus 10 close parenthesis squared equals 0")[$attach(( b + 10 ), t: 2) = 0$]

#math.equation(block: true, alt: "open parenthesis 2 x minus 1 close parenthesis squared equals 0")[$attach(( 2 x − 1 ), t: 2) = 0$]

#solutionbox[
#math.equation(block: true, alt: "x equals 1 / 2")[$x = 1 "/" 2$]
]

#math.equation(block: true, alt: "open parenthesis 3 y plus 5 close parenthesis squared equals 0")[$attach(( 3 y + 5 ), t: 2) = 0$]

#strong[Solve Quadratic Equations by Factoring]

In the following exercises, solve.

#math.equation(block: true, alt: "x squared plus 7 x plus 12 equals 0")[$x^(2) + 7 x + 12 = 0$]

#solutionbox[
#math.equation(block: true, alt: "x equals −3 , x equals −4")[$x = −3 , x = −4$]
]

#math.equation(block: true, alt: "y squared minus 8 y plus 15 equals 0")[$y^(2) − 8 y + 15 = 0$]

#math.equation(block: true, alt: "5 a squared minus 26 a equals 24")[$5 a^(2) − 26 a = 24$]

#solutionbox[
#math.equation(block: true, alt: "a equals −4 / 5 , a equals 6")[$a = −4 "/" 5 , a = 6$]
]

#math.equation(block: true, alt: "4 b squared plus 7 b equals −3")[$4 b^(2) + 7 b = −3$]

#math.equation(block: true, alt: "4 m squared equals 17 m minus 15")[$4 m^(2) = 17 m − 15$]

#solutionbox[
#math.equation(block: true, alt: "m equals 5 / 4 , m equals 3")[$m = 5 "/" 4 , m = 3$]
]

#math.equation(block: true, alt: "n squared equals 5 n minus 6")[$n^(2) = 5 n − 6$]

#math.equation(block: true, alt: "7 a squared plus 14 a equals 7 a")[$7 a^(2) + 14 a = 7 a$]

#solutionbox[
#math.equation(block: true, alt: "a equals −1 , a equals 0")[$a = −1 , a = 0$]
]

#math.equation(block: true, alt: "12 b squared minus 15 b equals −9 b")[$12 b^(2) − 15 b = −9 b$]

#math.equation(block: true, alt: "49 m squared equals 144")[$49 m^(2) = 144$]

#solutionbox[
#math.equation(block: true, alt: "m equals 12 / 7 , m equals −12 / 7")[$m = 12 "/" 7 , m = −12 "/" 7$]
]

#math.equation(block: true, alt: "625 equals x squared")[$625 = x^(2)$]

#math.equation(block: true, alt: "open parenthesis y minus 3 close parenthesis open parenthesis y plus 2 close parenthesis equals 4 y")[$( y − 3 ) ( y + 2 ) = 4 y$]

#solutionbox[
#math.equation(block: true, alt: "y equals −1 , y equals 6")[$y = −1 , y = 6$]
]

#math.equation(block: true, alt: "open parenthesis p minus 5 close parenthesis open parenthesis p plus 3 close parenthesis equals −7")[$( p − 5 ) ( p + 3 ) = −7$]

#math.equation(block: true, alt: "open parenthesis 2 x plus 1 close parenthesis open parenthesis x minus 3 close parenthesis equals −4 x")[$( 2 x + 1 ) ( x − 3 ) = −4 x$]

#solutionbox[
#math.equation(block: true, alt: "x equals 3 / 2 , x equals −1")[$x = 3 "/" 2 , x = −1$]
]

#math.equation(block: true, alt: "open parenthesis x plus 6 close parenthesis open parenthesis x minus 3 close parenthesis equals −8")[$( x + 6 ) ( x − 3 ) = −8$]

#math.equation(block: true, alt: "16 p cubed equals 24 p squared minus 9 p")[$16 p^(3) = 24 p^(2) − 9 p$]

#solutionbox[
#math.equation(block: true, alt: "p equals 0 , p equals ¾")[$p = 0 , p = ¾$]
]

#math.equation(block: true, alt: "m cubed minus 2 m squared equals − m")[$m^(3) − 2 m^(2) = "−" m$]

#math.equation(block: true, alt: "20 x squared minus 60 x equals −45")[$20 x^(2) − 60 x = −45$]

#solutionbox[
#math.equation(block: true, alt: "x equals 3 / 2")[$x = 3 "/" 2$]
]

#math.equation(block: true, alt: "3 y squared minus 18 y equals −27")[$3 y^(2) − 18 y = −27$]

#strong[Solve Applications Modeled by Quadratic Equations]

In the following exercises, solve.

The product of two consecutive integers is 56. Find the integers.

#solutionbox[
#math.equation(block: true, alt: "7 and 8 ; minus 8 and −7")[$7 #h(0.2em) "and" #h(0.2em) 8 ; − 8 #h(0.2em) "and" #h(0.2em) −7$]
]

The product of two consecutive integers is 42. Find the integers.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

#solutionbox[
#math.equation(block: true, alt: "4 feet and 7 feet")[$4 #h(0.2em) "feet and" #h(0.2em) 7 #h(0.2em) "feet"$]
]

A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

#solutionbox[
#math.equation(block: true, alt: "6 feet and 8 feet")[$6 #h(0.2em) "feet and" #h(0.2em) 8 #h(0.2em) "feet"$]
]

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

#strong[Mixed Practice]

In the following exercises, solve.

#math.equation(block: true, alt: "open parenthesis x plus 8 close parenthesis open parenthesis x minus 3 close parenthesis equals 0")[$( x + 8 ) ( x − 3 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "x equals −8 , x equals 3")[$x = −8 , x = 3$]
]

#math.equation(block: true, alt: "open parenthesis 3 y minus 5 close parenthesis open parenthesis y plus 7 close parenthesis equals 0")[$( 3 y − 5 ) ( y + 7 ) = 0$]

#math.equation(block: true, alt: "p squared plus 12 p plus 11 equals 0")[$p^(2) + 12 p + 11 = 0$]

#solutionbox[
#math.equation(block: true, alt: "p equals −1 , p equals −11")[$p = −1 , p = −11$]
]

#math.equation(block: true, alt: "q squared minus 12 q minus 13 equals 0")[$q^(2) − 12 q − 13 = 0$]

#math.equation(block: true, alt: "m squared equals 6 m plus 16")[$m^(2) = 6 m + 16$]

#solutionbox[
#math.equation(block: true, alt: "m equals −2 , m equals 8")[$m = −2 , m = 8$]
]

#math.equation(block: true, alt: "4 n squared plus 19 n equals 5")[$4 n^(2) + 19 n = 5$]

#math.equation(block: true, alt: "a cubed minus a squared minus 42 a equals 0")[$a^(3) − a^(2) − 42 a = 0$]

#solutionbox[
#math.equation(block: true, alt: "a equals 0 , a equals −6 , a equals 7")[$a = 0 , a = −6 , a = 7$]
]

#math.equation(block: true, alt: "4 b squared minus 60 b plus 224 equals 0")[$4 b^(2) − 60 b + 224 = 0$]

The product of two consecutive integers is 110. Find the integers.

#solutionbox[
#math.equation(block: true, alt: "10 and 11 ; minus 11 and −10")[$10 #h(0.2em) "and" #h(0.2em) 11 ; − 11 #h(0.2em) "and" #h(0.2em) −10$]
]

The length of one leg of a right triangle is three feet more than the other leg. If the hypotenuse is 15 feet, find the lengths of the two legs.


==== Everyday Math

#strong[Area of a patio] If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation #math.equation(block: false, alt: "open parenthesis s plus 4 close parenthesis squared equals 196")[$attach(( s + 4 ), t: 2) = 196$] for #emph[s] to find the length of a side of the patio.

#solutionbox[
10 feet
]

#strong[Watermelon drop] A watermelon is dropped from the tenth story of a building. Solve the equation #math.equation(block: false, alt: "−16 t squared plus 144 equals 0")[$−16 t^(2) + 144 = 0$] for #math.equation(block: false, alt: "t")[$t$] to find the number of seconds it takes the watermelon to reach the ground.


==== Writing Exercises

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

#solutionbox[
Answers may vary.
]

Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has the following statements all to be preceded by “I can…”. The first row is “solve quadratic equations by using the zero product property”. The second row is “solve quadratic equations by factoring”. The third row is “solve applications modeled by quadratic equations”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.], alt: "This table has the following statements all to be preceded by “I can…”. The first row is “solve quadratic equations by using the zero product property”. The second row is “solve quadratic equations by factoring”. The third row is “solve applications modeled by quadratic equations”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.", caption: none)

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


=== Chapter 7 Review Exercises


==== 7.1 Greatest Common Factor and Factor by Grouping

#strong[Find the Greatest Common Factor of Two or More Expressions]

In the following exercises, find the greatest common factor.

#math.equation(block: true, alt: "42 , 60")[$42 , 60$]

#solutionbox[
6
]

#math.equation(block: true, alt: "450 , 420")[$450 , 420$]

#math.equation(block: true, alt: "90 , 150 , 105")[$90 , 150 , 105$]

#solutionbox[
#math.equation(block: true, alt: "15")[$15$]
]

#math.equation(block: true, alt: "60 , 294 , 630")[$60 , 294 , 630$]

#strong[Factor the Greatest Common Factor from a Polynomial]

In the following exercises, factor the greatest common factor from each polynomial.

#math.equation(block: true, alt: "24 x minus 42")[$24 x − 42$]

#solutionbox[
#math.equation(block: true, alt: "6 open parenthesis 4 x minus 7 close parenthesis")[$6 ( 4 x − 7 )$]
]

#math.equation(block: true, alt: "35 y plus 84")[$35 y + 84$]

#math.equation(block: true, alt: "15 m to the power 4 plus 6 m squared n")[$15 m^(4) + 6 m^(2) n$]

#solutionbox[
#math.equation(block: true, alt: "3 m squared open parenthesis 5 m squared plus 2 n close parenthesis")[$3 m^(2) ( 5 m^(2) + 2 n )$]
]

#math.equation(block: true, alt: "24 p t to the power 4 plus 16 t to the power 7")[$24 p t^(4) + 16 t^(7)$]

#strong[Factor by Grouping]

In the following exercises, factor by grouping.

#math.equation(block: true, alt: "a x minus a y plus b x minus b y")[$a x − a y + b x − b y$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis open parenthesis x minus y close parenthesis")[$( a + b ) ( x − y )$]
]

#math.equation(block: true, alt: "x squared y minus x y squared plus 2 x minus 2 y")[$x^(2) y − x y^(2) + 2 x − 2 y$]

#math.equation(block: true, alt: "x squared plus 7 x minus 3 x minus 21")[$x^(2) + 7 x − 3 x − 21$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 7 close parenthesis")[$( x − 3 ) ( x + 7 )$]
]

#math.equation(block: true, alt: "4 x squared minus 16 x plus 3 x minus 12")[$4 x^(2) − 16 x + 3 x − 12$]

#math.equation(block: true, alt: "m cubed plus m squared plus m plus 1")[$m^(3) + m^(2) + m + 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis m squared plus 1 close parenthesis open parenthesis m plus 1 close parenthesis")[$( m^(2) + 1 ) ( m + 1 )$]
]

#math.equation(block: true, alt: "5 x minus 5 y minus y plus x")[$5 x − 5 y − y + x$]


==== 7.2 Factor Trinomials of the form #math.equation(block: false, alt: "x squared plus b x plus c")[$x^(2) + b x + c$]

#strong[Factor Trinomials of the Form #math.equation(block: false, alt: "x squared plus b x plus c")[$x^(2) + b x + c$]]

In the following exercises, factor each trinomial of the form #math.equation(block: false, alt: "x squared plus b x plus c")[$x^(2) + b x + c$].

#math.equation(block: true, alt: "u squared plus 17 u plus 72")[$u^(2) + 17 u + 72$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis u plus 8 close parenthesis open parenthesis u plus 9 close parenthesis")[$( u + 8 ) ( u + 9 )$]
]

#math.equation(block: true, alt: "a squared plus 14 a plus 33")[$a^(2) + 14 a + 33$]

#math.equation(block: true, alt: "k squared minus 16 k plus 60")[$k^(2) − 16 k + 60$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis k minus 6 close parenthesis open parenthesis k minus 10 close parenthesis")[$( k − 6 ) ( k − 10 )$]
]

#math.equation(block: true, alt: "r squared minus 11 r plus 28")[$r^(2) − 11 r + 28$]

#math.equation(block: true, alt: "y squared plus 6 y minus 7")[$y^(2) + 6 y − 7$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis y plus 7 close parenthesis open parenthesis y minus 1 close parenthesis")[$( y + 7 ) ( y − 1 )$]
]

#math.equation(block: true, alt: "m squared plus 3 m minus 54")[$m^(2) + 3 m − 54$]

#math.equation(block: true, alt: "s squared minus 2 s minus 8")[$s^(2) − 2 s − 8$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis s minus 4 close parenthesis open parenthesis s plus 2 close parenthesis")[$( s − 4 ) ( s + 2 )$]
]

#math.equation(block: true, alt: "x squared minus 3 x minus 10")[$x^(2) − 3 x − 10$]

#strong[Factor Trinomials of the Form #math.equation(block: false, alt: "x squared plus b x y plus c y squared")[$x^(2) + b x y + c y^(2)$]]

In the following examples, factor each trinomial of the form #math.equation(block: false, alt: "x squared plus b x y plus c y squared")[$x^(2) + b x y + c y^(2)$].

#math.equation(block: true, alt: "x squared plus 12 x y plus 35 y squared")[$x^(2) + 12 x y + 35 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 5 y close parenthesis open parenthesis x plus 7 y close parenthesis")[$( x + 5 y ) ( x + 7 y )$]
]

#math.equation(block: true, alt: "u squared plus 14 u v plus 48 v squared")[$u^(2) + 14 u v + 48 v^(2)$]

#math.equation(block: true, alt: "a squared plus 4 a b minus 21 b squared")[$a^(2) + 4 a b − 21 b^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a plus 7 b close parenthesis open parenthesis a minus 3 b close parenthesis")[$( a + 7 b ) ( a − 3 b )$]
]

#math.equation(block: true, alt: "p squared minus 5 p q minus 36 q squared")[$p^(2) − 5 p q − 36 q^(2)$]


==== 7.3 Factoring Trinomials of the form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$]

#strong[Recognize a Preliminary Strategy to Factor Polynomials Completely]

In the following exercises, identify the best method to use to factor each polynomial.

#math.equation(block: true, alt: "y squared minus 17 y plus 42")[$y^(2) − 17 y + 42$]

#solutionbox[
Undo FOIL
]

#math.equation(block: true, alt: "12 r squared plus 32 r plus 5")[$12 r^(2) + 32 r + 5$]

#math.equation(block: true, alt: "8 a cubed plus 72 a")[$8 a^(3) + 72 a$]

#solutionbox[
Factor the GCF
]

#math.equation(block: true, alt: "4 m minus m n minus 3 n plus 12")[$4 m − m n − 3 n + 12$]

#strong[Factor Trinomials of the Form #math.equation(block: false, alt: "a x squared plus b x plus c")[$a x^(2) + b x + c$] with a GCF]

In the following exercises, factor completely.

#math.equation(block: true, alt: "6 x squared plus 42 x plus 60")[$6 x^(2) + 42 x + 60$]

#solutionbox[
#math.equation(block: true, alt: "6 open parenthesis x plus 2 close parenthesis open parenthesis x plus 5 close parenthesis")[$6 ( x + 2 ) ( x + 5 )$]
]

#math.equation(block: true, alt: "8 a squared plus 32 a plus 24")[$8 a^(2) + 32 a + 24$]

#math.equation(block: true, alt: "3 n to the power 4 minus 12 n cubed minus 96 n squared")[$3 n^(4) − 12 n^(3) − 96 n^(2)$]

#solutionbox[
#math.equation(block: true, alt: "3 n squared open parenthesis n minus 8 close parenthesis open parenthesis n plus 4 close parenthesis")[$3 n^(2) ( n − 8 ) ( n + 4 )$]
]

#math.equation(block: true, alt: "5 y cubed plus 25 y squared minus 70 y")[$5 y^(3) + 25 y^(2) − 70 y$]

#strong[Factor Trinomials Using the “ac” Method]

In the following exercises, factor.

#math.equation(block: true, alt: "2 x squared plus 9 x plus 4")[$2 x^(2) + 9 x + 4$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 4 close parenthesis open parenthesis 2 x plus 1 close parenthesis")[$( x + 4 ) ( 2 x + 1 )$]
]

#math.equation(block: true, alt: "3 y squared plus 17 y plus 10")[$3 y^(2) + 17 y + 10$]

#math.equation(block: true, alt: "18 a squared minus 9 a plus 1")[$18 a^(2) − 9 a + 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 a minus 1 close parenthesis open parenthesis 6 a minus 1 close parenthesis")[$( 3 a − 1 ) ( 6 a − 1 )$]
]

#math.equation(block: true, alt: "8 u squared minus 14 u plus 3")[$8 u^(2) − 14 u + 3$]

#math.equation(block: true, alt: "15 p squared plus 2 p minus 8")[$15 p^(2) + 2 p − 8$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 p plus 4 close parenthesis open parenthesis 3 p minus 2 close parenthesis")[$( 5 p + 4 ) ( 3 p − 2 )$]
]

#math.equation(block: true, alt: "15 x squared plus x minus 2")[$15 x^(2) + x − 2$]

#math.equation(block: true, alt: "40 s squared minus s minus 6")[$40 s^(2) − s − 6$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 s minus 2 close parenthesis open parenthesis 8 s plus 3 close parenthesis")[$( 5 s − 2 ) ( 8 s + 3 )$]
]

#math.equation(block: true, alt: "20 n squared minus 7 n minus 3")[$20 n^(2) − 7 n − 3$]

#strong[Factor Trinomials with a GCF Using the “ac” Method]

In the following exercises, factor.

#math.equation(block: true, alt: "3 x squared plus 3 x minus 36")[$3 x^(2) + 3 x − 36$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis x plus 4 close parenthesis open parenthesis x minus 3 close parenthesis")[$3 ( x + 4 ) ( x − 3 )$]
]

#math.equation(block: true, alt: "4 x squared plus 4 x minus 8")[$4 x^(2) + 4 x − 8$]

#math.equation(block: true, alt: "60 y squared minus 85 y minus 25")[$60 y^(2) − 85 y − 25$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis 4 y plus 1 close parenthesis open parenthesis 3 y minus 5 close parenthesis")[$5 ( 4 y + 1 ) ( 3 y − 5 )$]
]

#math.equation(block: true, alt: "18 a squared minus 57 a minus 21")[$18 a^(2) − 57 a − 21$]


==== 7.4 Factoring Special Products

#strong[Factor Perfect Square Trinomials]

In the following exercises, factor.

#math.equation(block: true, alt: "25 x squared plus 30 x plus 9")[$25 x^(2) + 30 x + 9$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 x plus 3 close parenthesis squared")[$attach(( 5 x + 3 ), t: 2)$]
]

#math.equation(block: true, alt: "16 y squared plus 72 y plus 81")[$16 y^(2) + 72 y + 81$]

#math.equation(block: true, alt: "36 a squared minus 84 a b plus 49 b squared")[$36 a^(2) − 84 a b + 49 b^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 6 a minus 7 b close parenthesis squared")[$attach(( 6 a − 7 b ), t: 2)$]
]

#math.equation(block: true, alt: "64 r squared minus 176 r s plus 121 s squared")[$64 r^(2) − 176 r s + 121 s^(2)$]

#math.equation(block: true, alt: "40 x squared plus 360 x plus 810")[$40 x^(2) + 360 x + 810$]

#solutionbox[
#math.equation(block: true, alt: "10 open parenthesis 2 x plus 9 close parenthesis squared")[$10 attach(( 2 x + 9 ), t: 2)$]
]

#math.equation(block: true, alt: "75 u squared plus 180 u plus 108")[$75 u^(2) + 180 u + 108$]

#math.equation(block: true, alt: "2 y cubed minus 16 y squared plus 32 y")[$2 y^(3) − 16 y^(2) + 32 y$]

#solutionbox[
#math.equation(block: true, alt: "2 y open parenthesis y minus 4 close parenthesis squared")[$2 y attach(( y − 4 ), t: 2)$]
]

#math.equation(block: true, alt: "5 k cubed minus 70 k squared plus 245 k")[$5 k^(3) − 70 k^(2) + 245 k$]

#strong[Factor Differences of Squares]

In the following exercises, factor.

#math.equation(block: true, alt: "81 r squared minus 25")[$81 r^(2) − 25$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 9 r minus 5 close parenthesis open parenthesis 9 r plus 5 close parenthesis")[$( 9 r − 5 ) ( 9 r + 5 )$]
]

#math.equation(block: true, alt: "49 a squared minus 144")[$49 a^(2) − 144$]

#math.equation(block: true, alt: "169 m squared minus n squared")[$169 m^(2) − n^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 13 m plus n close parenthesis open parenthesis 13 m minus n close parenthesis")[$( 13 m + n ) ( 13 m − n )$]
]

#math.equation(block: true, alt: "64 x squared minus y squared")[$64 x^(2) − y^(2)$]

#math.equation(block: true, alt: "25 p squared minus 1")[$25 p^(2) − 1$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 5 p minus 1 close parenthesis open parenthesis 5 p plus 1 close parenthesis")[$( 5 p − 1 ) ( 5 p + 1 )$]
]

#math.equation(block: true, alt: "1 minus 16 s squared")[$1 − 16 s^(2)$]

#math.equation(block: true, alt: "9 minus 121 y squared")[$9 − 121 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 plus 11 y close parenthesis open parenthesis 3 minus 11 y close parenthesis")[$( 3 + 11 y ) ( 3 − 11 y )$]
]

#math.equation(block: true, alt: "100 k squared minus 81")[$100 k^(2) − 81$]

#math.equation(block: true, alt: "20 x squared minus 125")[$20 x^(2) − 125$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis 2 x minus 5 close parenthesis open parenthesis 2 x plus 5 close parenthesis")[$5 ( 2 x − 5 ) ( 2 x + 5 )$]
]

#math.equation(block: true, alt: "18 y squared minus 98")[$18 y^(2) − 98$]

#math.equation(block: true, alt: "49 u cubed minus 9 u")[$49 u^(3) − 9 u$]

#solutionbox[
#math.equation(block: true, alt: "u open parenthesis 7 u plus 3 close parenthesis open parenthesis 7 u minus 3 close parenthesis")[$u ( 7 u + 3 ) ( 7 u − 3 )$]
]

#math.equation(block: true, alt: "169 n cubed minus n")[$169 n^(3) − n$]

#strong[Factor Sums and Differences of Cubes]

In the following exercises, factor.

#math.equation(block: true, alt: "a cubed minus 125")[$a^(3) − 125$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a minus 5 close parenthesis open parenthesis a squared plus 5 a plus 25 close parenthesis")[$( a − 5 ) ( a^(2) + 5 a + 25 )$]
]

#math.equation(block: true, alt: "b cubed minus 216")[$b^(3) − 216$]

#math.equation(block: true, alt: "2 m cubed plus 54")[$2 m^(3) + 54$]

#solutionbox[
#math.equation(block: true, alt: "2 open parenthesis m plus 3 close parenthesis open parenthesis m squared minus 3 m plus 9 close parenthesis")[$2 ( m + 3 ) ( m^(2) − 3 m + 9 )$]
]

#math.equation(block: true, alt: "81 x cubed plus 3")[$81 x^(3) + 3$]


==== 7.5 General Strategy for Factoring Polynomials

#strong[Recognize and Use the Appropriate Method to Factor a Polynomial Completely]

In the following exercises, factor completely.

#math.equation(block: true, alt: "24 x cubed plus 44 x squared")[$24 x^(3) + 44 x^(2)$]

#solutionbox[
#math.equation(block: true, alt: "4 x squared open parenthesis 6 x plus 11 close parenthesis")[$4 x^(2) ( 6 x + 11 )$]
]

#math.equation(block: true, alt: "24 a to the power 4 minus 9 a cubed")[$24 a^(4) − 9 a^(3)$]

#math.equation(block: true, alt: "16 n squared minus 56 m n plus 49 m squared")[$16 n^(2) − 56 m n + 49 m^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 n minus 7 m close parenthesis squared")[$attach(( 4 n − 7 m ), t: 2)$]
]

#math.equation(block: true, alt: "6 a squared minus 25 a minus 9")[$6 a^(2) − 25 a − 9$]

#math.equation(block: true, alt: "5 r squared plus 22 r minus 48")[$5 r^(2) + 22 r − 48$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis r plus 6 close parenthesis open parenthesis 5 r minus 8 close parenthesis")[$( r + 6 ) ( 5 r − 8 )$]
]

#math.equation(block: true, alt: "5 u to the power 4 minus 45 u squared")[$5 u^(4) − 45 u^(2)$]

#math.equation(block: true, alt: "n to the power 4 minus 81")[$n^(4) − 81$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis n squared plus 9 close parenthesis open parenthesis n plus 3 close parenthesis open parenthesis n minus 3 close parenthesis")[$( n^(2) + 9 ) ( n + 3 ) ( n − 3 )$]
]

#math.equation(block: true, alt: "64 j squared plus 225")[$64 j^(2) + 225$]

#math.equation(block: true, alt: "5 x squared plus 5 x minus 60")[$5 x^(2) + 5 x − 60$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis x minus 3 close parenthesis open parenthesis x plus 4 close parenthesis")[$5 ( x − 3 ) ( x + 4 )$]
]

#math.equation(block: true, alt: "b cubed minus 64")[$b^(3) − 64$]

#math.equation(block: true, alt: "m cubed plus 125")[$m^(3) + 125$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis m plus 5 close parenthesis open parenthesis m squared minus 5 m plus 25 close parenthesis")[$( m + 5 ) ( m^(2) − 5 m + 25 )$]
]

#math.equation(block: true, alt: "2 b squared minus 2 b c plus 5 c b minus 5 c squared")[$2 b^(2) − 2 b c + 5 c b − 5 c^(2)$]


==== 7.6 Quadratic Equations

#strong[Use the Zero Product Property]

In the following exercises, solve.

#math.equation(block: true, alt: "open parenthesis a minus 3 close parenthesis open parenthesis a plus 7 close parenthesis equals 0")[$( a − 3 ) ( a + 7 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "a equals 3 , a equals −7")[$a = 3 , #h(0.2em) a = −7$]
]

#math.equation(block: true, alt: "open parenthesis b minus 3 close parenthesis open parenthesis b plus 10 close parenthesis equals 0")[$( b − 3 ) ( b + 10 ) = 0$]

#math.equation(block: true, alt: "3 m open parenthesis 2 m minus 5 close parenthesis open parenthesis m plus 6 close parenthesis equals 0")[$3 m ( 2 m − 5 ) ( m + 6 ) = 0$]

#solutionbox[
#math.equation(block: true, alt: "m equals 0 m equals –6 m equals the fraction 5 over 2")[$m = 0 #h(0.2em) m = –6 #h(0.2em) m = frac(5, 2)$]
]

#math.equation(block: true, alt: "7 n open parenthesis 3 n plus 8 close parenthesis open parenthesis n minus 5 close parenthesis equals 0")[$7 n ( 3 n + 8 ) ( n − 5 ) = 0$]

#strong[Solve Quadratic Equations by Factoring]

In the following exercises, solve.

#math.equation(block: true, alt: "x squared plus 9 x plus 20 equals 0")[$x^(2) + 9 x + 20 = 0$]

#solutionbox[
#math.equation(block: true, alt: "x equals −4 , x equals −5")[$x = −4 , x = −5$]
]

#math.equation(block: true, alt: "y squared minus y minus 72 equals 0")[$y^(2) − y − 72 = 0$]

#math.equation(block: true, alt: "2 p squared minus 11 p equals 40")[$2 p^(2) − 11 p = 40$]

#solutionbox[
#math.equation(block: true, alt: "p equals minus the fraction 5 over 2 , p equals 8")[$p = − frac(5, 2) , p = 8$]
]

#math.equation(block: true, alt: "q cubed plus 3 q squared plus 2 q equals 0")[$q^(3) + 3 q^(2) + 2 q = 0$]

#math.equation(block: true, alt: "144 m squared minus 25 equals 0")[$144 m^(2) − 25 = 0$]

#solutionbox[
#math.equation(block: true, alt: "m equals the fraction 5 over 12 , m equals minus the fraction 5 over 12")[$m = frac(5, 12) , m = − frac(5, 12)$]
]

#math.equation(block: true, alt: "4 n squared equals 36")[$4 n^(2) = 36$]

#strong[Solve Applications Modeled by Quadratic Equations]

In the following exercises, solve.

The product of two consecutive numbers is #math.equation(block: false, alt: "462")[$462$]. Find the numbers.

#solutionbox[
#math.equation(block: true, alt: "−21 and −22 ; 21 and 22")[$−21 #h(0.2em) "and" #h(0.2em) −22 ; #h(0.2em) 21 #h(0.2em) "and" #h(0.2em) 22$]
]

The area of a rectangular shaped patio #math.equation(block: false, alt: "400")[$400$] square feet. The length of the patio is #math.equation(block: false, alt: "9")[$9$] feet more than its width. Find the length and width.


=== Practice Test

In the following exercises, find the Greatest Common Factor in each expression.

#math.equation(block: true, alt: "14 y minus 42")[$14 y − 42$]

#solutionbox[
#math.equation(block: true, alt: "14 open parenthesis y minus 3 close parenthesis")[$14 ( y − 3 )$]
]

#math.equation(block: true, alt: "−6 x squared minus 30 x")[$−6 x^(2) − 30 x$]

#math.equation(block: true, alt: "80 a squared plus 120 a cubed")[$80 a^(2) + 120 a^(3)$]

#solutionbox[
#math.equation(block: true, alt: "40 a squared open parenthesis 2 plus 3 a close parenthesis")[$40 a^(2) ( 2 + 3 a )$]
]

#math.equation(block: true, alt: "5 m open parenthesis m minus 1 close parenthesis plus 3 open parenthesis m minus 1 close parenthesis")[$5 m ( m − 1 ) + 3 ( m − 1 )$]

In the following exercises, factor completely.

#math.equation(block: true, alt: "x squared plus 13 x plus 42")[$x^(2) + 13 x + 42$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x plus 7 close parenthesis open parenthesis x plus 6 close parenthesis")[$( x + 7 ) ( x + 6 )$]
]

#math.equation(block: true, alt: "p squared plus p q minus 12 q squared")[$p^(2) + p q − 12 q^(2)$]

#math.equation(block: true, alt: "3 a cubed minus 6 a squared minus 72 a")[$3 a^(3) − 6 a^(2) − 72 a$]

#solutionbox[
#math.equation(block: true, alt: "3 a open parenthesis a minus 6 close parenthesis open parenthesis a plus 4 close parenthesis")[$3 a ( a − 6 ) ( a + 4 )$]
]

#math.equation(block: true, alt: "s squared minus 25 s plus 84")[$s^(2) − 25 s + 84$]

#math.equation(block: true, alt: "5 n squared plus 30 n plus 45")[$5 n^(2) + 30 n + 45$]

#solutionbox[
#math.equation(block: true, alt: "5 open parenthesis n plus 3 close parenthesis squared")[$5 ( n + 3 )^(2)$]
]

#math.equation(block: true, alt: "64 y squared minus 49")[$64 y^(2) − 49$]

#math.equation(block: true, alt: "x y minus 8 y plus 7 x minus 56")[$x y − 8 y + 7 x − 56$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis x minus 8 close parenthesis open parenthesis y plus 7 close parenthesis")[$( x − 8 ) ( y + 7 )$]
]

#math.equation(block: true, alt: "40 r squared plus 810")[$40 r^(2) + 810$]

#math.equation(block: true, alt: "9 s squared minus 12 s plus 4")[$9 s^(2) − 12 s + 4$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 3 s minus 2 close parenthesis squared")[$attach(( 3 s − 2 ), t: 2)$]
]

#math.equation(block: true, alt: "n squared plus 12 n plus 36")[$n^(2) + 12 n + 36$]

#math.equation(block: true, alt: "100 minus a squared")[$100 − a^(2)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 10 minus a close parenthesis open parenthesis 10 plus a close parenthesis")[$( 10 − a ) ( 10 + a )$]
]

#math.equation(block: true, alt: "6 x squared minus 11 x minus 10")[$6 x^(2) − 11 x − 10$]

#math.equation(block: true, alt: "3 x squared minus 75 y squared")[$3 x^(2) − 75 y^(2)$]

#solutionbox[
#math.equation(block: true, alt: "3 open parenthesis x plus 5 y close parenthesis open parenthesis x minus 5 y close parenthesis")[$3 ( x + 5 y ) ( x − 5 y )$]
]

#math.equation(block: true, alt: "c cubed minus 1000 d cubed")[$c^(3) − 1000 d^(3)$]

#math.equation(block: true, alt: "a b minus 3 b minus 2 a plus 6")[$a b − 3 b − 2 a + 6$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis a minus 3 close parenthesis open parenthesis b minus 2 close parenthesis")[$( a − 3 ) ( b − 2 )$]
]

#math.equation(block: true, alt: "6 u squared plus 3 u minus 18")[$6 u^(2) + 3 u − 18$]

#math.equation(block: true, alt: "8 m squared plus 22 m plus 5")[$8 m^(2) + 22 m + 5$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 4 m plus 1 close parenthesis open parenthesis 2 m plus 5 close parenthesis")[$( 4 m + 1 ) ( 2 m + 5 )$]
]

In the following exercises, solve.

#math.equation(block: true, alt: "x squared plus 9 x plus 20 equals 0")[$x^(2) + 9 x + 20 = 0$]

#math.equation(block: true, alt: "y squared equals y plus 132")[$y^(2) = y + 132$]

#solutionbox[
#math.equation(block: true, alt: "y equals −11 , y equals 12")[$"y" = −11 , "y" = 12$]
]

#math.equation(block: true, alt: "5 a squared plus 26 a equals 24")[$5 a^(2) + 26 a = 24$]

#math.equation(block: true, alt: "9 b squared minus 9 equals 0")[$9 b^(2) − 9 = 0$]

#solutionbox[
#math.equation(block: true, alt: "b equals 1 , b equals −1")[$b = 1 , b = −1$]
]

#math.equation(block: true, alt: "16 minus m squared equals 0")[$16 − m^(2) = 0$]

#math.equation(block: true, alt: "4 n squared plus 19 n plus 21 equals 0")[$4 n^(2) + 19 n + 21 = 0$]

#solutionbox[
#math.equation(block: true, alt: "n equals minus the fraction 7 over 4 , n equals −3")[$n = − frac(7, 4) , n = −3$]
]

#math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 2 close parenthesis equals 6")[$( x − 3 ) ( x + 2 ) = 6$]

The product of two consecutive integers is 156. Find the integers.

#solutionbox[
#math.equation(block: true, alt: "12 and 13 ; minus 13 and −12")[$12 #h(0.2em) "and" #h(0.2em) 13 ; − 13 #h(0.2em) "and" #h(0.2em) −12$]
]

The area of a rectangular place mat is #math.equation(block: false, alt: "168")[$168$] square inches. Its length is two inches longer than the width. Find the length and width of the place mat.