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== 9.6#h(0.6em)Solve Equations with Square Roots

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

Simplify: ⓐ #math.equation(block: false, alt: "the square root of 9")[$sqrt(9)$] ⓑ #math.equation(block: false, alt: "9 squared")[$9^(2)$]. #linebreak() 
If you missed this problem, review  and .

#solutionbox[
ⓐ #math.equation(block: false, alt: "3")[$3$] ⓑ #math.equation(block: false, alt: "81")[$81$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "5 open parenthesis x plus 1 close parenthesis minus 4 equals 3 open parenthesis 2 x minus 7 close parenthesis")[$5 ( x + 1 ) − 4 = 3 ( 2 x − 7 )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "22")[$22$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "n squared minus 6 n plus 8 equals 0")[$n^(2) − 6 n + 8 = 0$]. #linebreak() 
If you missed this problem, review .

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


=== Solve Radical Equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.

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

An equation in which the variable is in the radicand of a square root is called a #strong[radical equation].
]

As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.

But remember that when we write #math.equation(block: false, alt: "the square root of a")[$sqrt(a)$] we mean the principal square root. So #math.equation(block: false, alt: "the square root of a greater than or equal to 0")[$sqrt(a) ≥ 0$] always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make #math.equation(block: false, alt: "the square root of a")[$sqrt(a)$] negative. This algebraic solution would not be a solution to the original #strong[radical equation]; it is an #emph[extraneous solution.] We saw extraneous solutions when we solved rational equations, too.

#examplebox("Example 1")[][
For the equation #math.equation(block: false, alt: "the square root of x plus 2 equals x")[$sqrt(x + 2) = x$]:

ⓐ Is #math.equation(block: false, alt: "x equals 2")[$x = 2$] a solution? ⓑ Is #math.equation(block: false, alt: "x equals −1")[$x = −1$] a solution?

#solutionbox[
ⓐ Is #math.equation(block: false, alt: "x equals 2")[$x = 2$] a solution? #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation showing the square root of x plus 2 equals x, written as sqrt(x+2) = x.], alt: "A mathematical equation showing the square root of x plus 2 equals x, written as sqrt(x+2) = x.", caption: none)]),
  [Let #emph[x] = 2.], [#figure(figph[A mathematical expression: sqrt(2 + 2) =? 2. The solution is true, as sqrt(4) equals 2. The red '2's emphasize the numbers in question.], alt: "A mathematical expression: sqrt(2 + 2) =? 2. The solution is true, as sqrt(4) equals 2. The red '2's emphasize the numbers in question.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation questions whether the square root of 4 is equal to 2, represented as ''sqrt(4) ?= 2''. The question mark over the equals sign implies an inquiry into the truth of the statement.], alt: "A mathematical equation questions whether the square root of 4 is equal to 2, represented as ''sqrt(4) ?= 2''. The question mark over the equals sign implies an inquiry into the truth of the statement.", caption: none)],
  [], [#figure(figph[A simple mathematical equation '2 = 2' is shown, followed by a checkmark, indicating that the statement is correct.], alt: "A simple mathematical equation '2 = 2' is shown, followed by a checkmark, indicating that the statement is correct.", caption: none)],
  [], [2 is a solution.],
))

#linebreak() 
ⓑ Is #math.equation(block: false, alt: "x equals −1")[$x = −1$] a solution? #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing the square root of x plus 2 equals x. This algebraic problem involves a radical expression that needs to be solved for the variable x.], alt: "A mathematical equation is displayed, showing the square root of x plus 2 equals x. This algebraic problem involves a radical expression that needs to be solved for the variable x.", caption: none)]),
  [Let #emph[x] = −1.], [#figure(figph[Adding two to imaginary problems and still questioning if the result is negative. A complex math conundrum!], alt: "Adding two to imaginary problems and still questioning if the result is negative. A complex math conundrum!", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression shows the square root of 1 questioned as being equal to -1, which is generally incorrect as the principal square root of 1 is 1.], alt: "A mathematical expression shows the square root of 1 questioned as being equal to -1, which is generally incorrect as the principal square root of 1 is 1.", caption: none)],
  [], [#figure(figph[A mathematical expression displaying '1   -1' where the equals sign is struck through, indicating '1 is not equal to -1' on a white background.], alt: "A mathematical expression displaying '1 -1' where the equals sign is struck through, indicating '1 is not equal to -1' on a white background.", caption: none)],
  [], [−1 is not a solution.],
  [], [−1 is an extraneous solution to the equation.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the equation #math.equation(block: false, alt: "the square root of x plus 6 equals x")[$sqrt(x + 6) = x$]:

ⓐ Is #math.equation(block: false, alt: "x equals −2")[$x = −2$] a solution? ⓑ Is #math.equation(block: false, alt: "x equals 3")[$x = 3$] a solution?

#solutionbox[
ⓐ no ⓑ #math.equation(block: false, alt: "yes")[$"yes"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
For the equation #math.equation(block: false, alt: "the square root of − x plus 2 equals x")[$sqrt("−" x + 2) = x$]:

ⓐ Is #math.equation(block: false, alt: "x equals −2")[$x = −2$] a solution? ⓑ Is #math.equation(block: false, alt: "x equals 1")[$x = 1$] a solution?

#solutionbox[
ⓐ no ⓑ #math.equation(block: false, alt: "yes")[$"yes"$]
]
]

Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.

#math.equation(block: true, alt: "For a greater than or equal to 0 , open parenthesis the square root of a close parenthesis squared equals a")[$"For" #h(0.2em) a ≥ 0 , #h(0.5em) attach(( sqrt(a) ), t: 2) = a$]#examplebox("Example 2")[How to Solve Radical Equations][
Solve: #math.equation(block: false, alt: "the square root of 2 x minus 1 equals 7")[$sqrt(2 x − 1) = 7$].

#solutionbox[
#figure(figph[This table has three columns and four rows. The first row says, “Step 1. Isolate the radical on one side of equation. The square root of (2x minus 1) is already isolated on the left side.” It then shows the equation: the square root of (2x minus 1) equals 7.], alt: "This table has three columns and four rows. The first row says, “Step 1. Isolate the radical on one side of equation. The square root of (2x minus 1) is already isolated on the left side.” It then shows the equation: the square root of (2x minus 1) equals 7.", caption: none)

#figure(figph[The second row says, “Step 2. Square both sides of the equation. Remember, the square root of a squared equals a.” It then shows the equation: the square root of (2x minus 1) squared equals 7 squared.], alt: "The second row says, “Step 2. Square both sides of the equation. Remember, the square root of a squared equals a.” It then shows the equation: the square root of (2x minus 1) squared equals 7 squared.", caption: none)

#figure(figph[The third row then says, “Step 3. Solve the new equation.” It indicates that 2x minus 1 equals 49 or 2x equals 50 which means that x equals 25.], alt: "The third row then says, “Step 3. Solve the new equation.” It indicates that 2x minus 1 equals 49 or 2x equals 50 which means that x equals 25.", caption: none)

#figure(figph[The fourth row says, “Step 4. Check the answer. Check:” It then indicates the square root of (2x minus 1) equals 7. This becomes the square root of (2 times 25 minus 1) equals 7. This becomes the square root of (50 minus 1) equals 7. This becomes the square root of 49 equals 7, and thus 7 equals 7. The figure then states, “The solutions is x equals 25.”], alt: "The fourth row says, “Step 4. Check the answer. Check:” It then indicates the square root of (2x minus 1) equals 7. This becomes the square root of (2 times 25 minus 1) equals 7. This becomes the square root of (50 minus 1) equals 7. This becomes the square root of 49 equals 7, and thus 7 equals 7. The figure then states, “The solutions is x equals 25.”", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 3 x minus 5 equals 5")[$sqrt(3 x − 5) = 5$].

#solutionbox[
#math.equation(block: true, alt: "10")[$10$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 4 x plus 8 equals 6")[$sqrt(4 x + 8) = 6$].

#solutionbox[
#math.equation(block: true, alt: "7")[$7$]
]
]

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

+ Isolate the radical on one side of the equation.
+ Square both sides of the equation.
+ Solve the new equation.
+ Check the answer.
]

#examplebox("Example 3")[][
Solve: #math.equation(block: false, alt: "the square root of 5 n minus 4 minus 9 equals 0")[$sqrt(5 n − 4) − 9 = 0$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation shows the square root of 5n minus 4, with 9 then subtracted, all equaling 0: sqrt(5n - 4) - 9 = 0.], alt: "A mathematical equation shows the square root of 5n minus 4, with 9 then subtracted, all equaling 0: sqrt(5n - 4) - 9 = 0.", caption: none)]),
  [To isolate the radical, add 9 to both sides.], [#figure(figph[A mathematical equation shows a step in solving for 'n': 'the square root of 5n minus 4, minus 9 plus 9 equals 0 plus 9'. The red plus signs highlight the addition of 9 to both sides of the equation.], alt: "A mathematical equation shows a step in solving for 'n': 'the square root of 5n minus 4, minus 9 plus 9 equals 0 plus 9'. The red plus signs highlight the addition of 9 to both sides of the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation displays the square root of 5n minus 4, which is equal to 9. The expression 5n - 4 is entirely under the radical sign.], alt: "A mathematical equation displays the square root of 5n minus 4, which is equal to 9. The expression 5n - 4 is entirely under the radical sign.", caption: none)],
  [Square both sides of the equation.], [#figure(figph[A mathematical equation is shown where the square of the square root of (5n minus 4) is equal to the square of 9. The equation reads as (sqrt(5n-4))^2 = (9)^2.], alt: "A mathematical equation is shown where the square of the square root of (5n minus 4) is equal to the square of 9. The equation reads as (sqrt(5n-4))^2 = (9)^2.", caption: none)],
  [Solve the new equation.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '5n - 4 = 81'.], alt: "A mathematical equation is displayed on a white background, which reads '5n - 4 = 81'.", caption: none)],
  [], [#figure(figph[A mathematical equation is displayed on a white background, reading '5n = 85' in black font.], alt: "A mathematical equation is displayed on a white background, reading '5n = 85' in black font.", caption: none)],
  [], [#figure(figph[The text 'n = 17' is displayed on a white background.], alt: "The text 'n = 17' is displayed on a white background.", caption: none)],
  [Check the answer.], [],
  [#figure(figph[Mathematical steps demonstrating the verification of a solution for the radical equation sqrt(5n - 4) - 9 = 0, showing that n=17 satisfies the equation.], alt: "Mathematical steps demonstrating the verification of a solution for the radical equation sqrt(5n - 4) - 9 = 0, showing that n=17 satisfies the equation.", caption: none)], [],
  [], [The solution is #emph[n] = 17.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 3 m plus 2 minus 5 equals 0")[$sqrt(3 m + 2) − 5 = 0$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 23 over 3")[$frac(23, 3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 10 z plus 1 minus 2 equals 0")[$sqrt(10 z + 1) − 2 = 0$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 10")[$frac(3, 10)$]
]
]

#examplebox("Example 4")[][
Solve: #math.equation(block: false, alt: "the square root of 3 y plus 5 plus 2 equals 5")[$sqrt(3 y + 5) + 2 = 5$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed on a white background: the square root of (3y + 5) + 2 = 5.], alt: "A mathematical equation is displayed on a white background: the square root of (3y + 5) + 2 = 5.", caption: none)]),
  [To isolate the radical, subtract 2 from both sides.], [#figure(figph[An algebraic equation: sqrt(3y + 5) + 2 - 2 = 5 - 2. The numbers '-2' on both sides are highlighted in red, illustrating the subtraction of 2 to balance the equation.], alt: "An algebraic equation: sqrt(3y + 5) + 2 - 2 = 5 - 2. The numbers '-2' on both sides are highlighted in red, illustrating the subtraction of 2 to balance the equation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation showing the square root of 3y + 5 equals 3.], alt: "A mathematical equation showing the square root of 3y + 5 equals 3.", caption: none)],
  [Square both sides of the equation.], [#figure(figph[A mathematical equation shows (square root of 3y + 5) squared equals 3 squared.], alt: "A mathematical equation shows (square root of 3y + 5) squared equals 3 squared.", caption: none)],
  [Solve the new equation.], [#figure(figph[A close-up view of the linear equation 3y + 5 = 9, displayed in black text on a white background.], alt: "A close-up view of the linear equation 3y + 5 = 9, displayed in black text on a white background.", caption: none)],
  [], [#figure(figph[The image displays a mathematical equation, '3y = 4', written in a clean and clear font against a white background.], alt: "The image displays a mathematical equation, '3y = 4', written in a clean and clear font against a white background.", caption: none)],
  [], [#figure(figph[The image shows the mathematical equation y = 4/3, displayed against a plain white background.], alt: "The image shows the mathematical equation y = 4/3, displayed against a plain white background.", caption: none)],
  [Check the answer.], [],
  [#figure(figph[Checking the solution for a radical equation. Substituting y=4/3 into sqrt(3y+5)+2=5 shows that 5=5, confirming its validity.], alt: "Checking the solution for a radical equation. Substituting y=4/3 into sqrt(3y+5)+2=5 shows that 5=5, confirming its validity.", caption: none)], [],
  [], [The solution is #math.equation(block: false, alt: "y equals the fraction 4 over 3")[$y = frac(4, 3)$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 3 p plus 3 plus 3 equals 5")[$sqrt(3 p + 3) + 3 = 5$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 5 q plus 1 plus 4 equals 6")[$sqrt(5 q + 1) + 4 = 6$].

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

When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.

#examplebox("Example 5")[][
Solve: #math.equation(block: false, alt: "the square root of 9 k minus 2 plus 1 equals 0")[$sqrt(9 k − 2) + 1 = 0$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation showing the square root of (9k - 2), plus 1, equals 0. The equation is written as: sqrt(9k-2) + 1 = 0.], alt: "A mathematical equation showing the square root of (9k - 2), plus 1, equals 0. The equation is written as: sqrt(9k-2) + 1 = 0.", caption: none)]),
  [To isolate the radical, subtract 1 from both sides.], [#figure(figph[A mathematical equation shows the square root of '9k-2' plus one, minus one, which equals zero minus one. The subtractions of '1' on both sides of the equation are highlighted in red.], alt: "A mathematical equation shows the square root of '9k-2' plus one, minus one, which equals zero minus one. The subtractions of '1' on both sides of the equation are highlighted in red.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation shows the square root of 9k minus 2 equals negative 1.], alt: "A mathematical equation shows the square root of 9k minus 2 equals negative 1.", caption: none)],
  [Since the square root is equal to a negative number, the equation has no solution.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 2 r minus 3 plus 5 equals 0")[$sqrt(2 r − 3) + 5 = 0$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 7 s minus 3 plus 2 equals 0")[$sqrt(7 s − 3) + 2 = 0$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]

If one side of the equation is a binomial, we use the binomial squares formula when we square it.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Binomial Squares]

#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared, open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) &  &  &  &  & attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2)$]
]

Don’t forget the middle term!

#examplebox("Example 6")[][
Solve: #math.equation(block: false, alt: "the square root of p minus 1 plus 1 equals p")[$sqrt(p − 1) + 1 = p$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[A mathematical equation is displayed, showing the square root of (p-1) plus 1 equals p. The equation is represented as '×p-1 + 1 = p'.], alt: "A mathematical equation is displayed, showing the square root of (p-1) plus 1 equals p. The equation is represented as '×p-1 + 1 = p'.", caption: none)]),
  [To isolate the radical, subtract 1 from both sides.], [#figure(figph[A mathematical equation shows 'sqrt(p-1) + 1 - 1 = p-1'. The '-1' terms are highlighted in red, indicating cancellation or a specific step in a calculation.], alt: "A mathematical equation shows 'sqrt(p-1) + 1 - 1 = p-1'. The '-1' terms are highlighted in red, indicating cancellation or a specific step in a calculation.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation showing the square root of (p-1) equals (p-1).], alt: "A mathematical equation showing the square root of (p-1) equals (p-1).", caption: none)],
  [Square both sides of the equation.], [#figure(figph[A mathematical equation is displayed on a white background: (sqrt(p-1))^2 = (p-1)^2.], alt: "A mathematical equation is displayed on a white background: (sqrt(p-1))^2 = (p-1)^2.", caption: none)],
  [Simplify, then solve the new equation.], [#figure(figph[The algebraic equation p - 1 = p^2 - 2p + 1 is displayed on a white background.], alt: "The algebraic equation p - 1 = p^2 - 2p + 1 is displayed on a white background.", caption: none)],
  [It is a quadratic equation, so get zero on one side.], [#figure(figph[A quadratic equation, 0 = p^2 - 3p + 2, is displayed in black text on a white background.], alt: "A quadratic equation, 0 = p^2 - 3p + 2, is displayed in black text on a white background.", caption: none)],
  [Factor the right side.], [#figure(figph[A mathematical equation on a white background reads '0 = (p - 1)(p - 2)'.], alt: "A mathematical equation on a white background reads '0 = (p - 1)(p - 2)'.", caption: none)],
  [Use the zero product property.], [#figure(figph[Two mathematical equations are displayed horizontally: 0 = p - 1 followed by 0 = p - 2. The equations use the variable 'p' and integers.], alt: "Two mathematical equations are displayed horizontally: 0 = p - 1 followed by 0 = p - 2. The equations use the variable 'p' and integers.", caption: none)],
  [Solve each equation.], [#figure(figph[The image displays the text 'p = 1 p = 2' in black characters against a plain white background, likely indicating two distinct parameter values or conditions.], alt: "The image displays the text 'p = 1 p = 2' in black characters against a plain white background, likely indicating two distinct parameter values or conditions.", caption: none)],
  [Check the answers.], [],
  [#figure(figph[Mathematical proof demonstrating the equation sqrt(p-1) + 1 = p holds true for p=1 and p=2, showing detailed step-by-step verification for each value of p.], alt: "Mathematical proof demonstrating the equation sqrt(p-1) + 1 = p holds true for p=1 and p=2, showing detailed step-by-step verification for each value of p.", caption: none)], [],
  [], [The solutions are #emph[p] = 1, #emph[p] = 2.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of x minus 2 plus 2 equals x")[$sqrt(x − 2) + 2 = x$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of y minus 5 plus 5 equals y")[$sqrt(y − 5) + 5 = y$].

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

#examplebox("Example 7")[][
Solve: #math.equation(block: false, alt: "the square root of r plus 4 minus r plus 2 equals 0")[$sqrt(r + 4) − r + 2 = 0$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of r plus 4 minus r plus 2 equals 0")[$sqrt(r + 4) − r + 2 = 0$]]),
  [Isolate the radical.], [#math.equation(block: false, alt: "the square root of r plus 4 equals r minus 2")[$#h(3.17em) sqrt(r + 4) = r − 2$]],
  [Square both sides of the equation.], [#math.equation(block: false, alt: "open parenthesis the square root of r plus 4 close parenthesis squared equals open parenthesis r minus 2 close parenthesis squared")[$#h(2.26em) attach(( sqrt(r + 4) ), t: 2) = attach(( r − 2 ), t: 2)$]],
  [Solve the new equation.], [#math.equation(block: false, alt: "r plus 4 equals r squared minus 4 r plus 4")[$#h(3.63em) r + 4 = r^(2) − 4 r + 4$]],
  [It is a quadratic equation, so get zero on one side.], [#math.equation(block: false, alt: "0 equals r squared minus 5 r")[$#h(5.17em) 0 = r^(2) − 5 r$]],
  [Factor the right side.], [#math.equation(block: false, alt: "0 equals r open parenthesis r minus 5 close parenthesis")[$#h(5.17em) 0 = r ( r − 5 )$]],
  [Use the zero product property.], [#math.equation(block: false, alt: "0 equals r 0 equals r minus 5")[$#h(1.99em) 0 = r #h(1em) 0 = r − 5$]],
  [Solve the equation.], [#math.equation(block: false, alt: "r equals 0 r equals 5")[$#h(2.09em) r = 0 #h(1em) r = 5$]],
  [Check the answer.], [],
  [#figure(figph[This image illustrates checking two potential solutions for the equation sqrt(r+4) - r + 2 = 0. It shows that r=0 is not a solution, but r=5 correctly satisfies the equation.], alt: "This image illustrates checking two potential solutions for the equation sqrt(r+4) - r + 2 = 0. It shows that r=0 is not a solution, but r=5 correctly satisfies the equation.", caption: none)], [The solution is #math.equation(block: false, alt: "r equals 5")[$r = 5$].],
  [], [#math.equation(block: false, alt: "r equals 0")[$r = 0$] is an extraneous solution.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of m plus 9 minus m plus 3 equals 0")[$sqrt(m + 9) − m + 3 = 0$].

#solutionbox[
#math.equation(block: true, alt: "7")[$7$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of n plus 1 minus n plus 1 equals 0")[$sqrt(n + 1) − n + 1 = 0$].

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

When there is a coefficient in front of the radical, we must square it, too.

#examplebox("Example 8")[][
Solve: #math.equation(block: false, alt: "3 the square root of 3 x minus 5 minus 8 equals 4")[$3 sqrt(3 x − 5) − 8 = 4$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the square root of 3 x minus 5 minus 8 equals 4")[$3 sqrt(3 x − 5) − 8 = 4$]]),
  [Isolate the radical.], [#math.equation(block: false, alt: "3 the square root of 3 x minus 5 equals 12")[$#h(1.62em) 3 sqrt(3 x − 5) = 12$]],
  [Square both sides of the equation.], [#math.equation(block: false, alt: "open parenthesis 3 the square root of 3 x minus 5 close parenthesis squared equals open parenthesis 12 close parenthesis squared")[$#h(0.67em) attach(( 3 sqrt(3 x − 5) ), t: 2) = attach(( 12 ), t: 2)$]],
  [Simplify, then solve the new equation.], [#math.equation(block: false, alt: "9 open parenthesis 3 x minus 5 close parenthesis equals 144")[$#h(1.47em) 9 ( 3 x − 5 ) = 144$]],
  [Distribute.], [#math.equation(block: false, alt: "27 x minus 45 equals 144")[$#h(1.59em) 27 x − 45 = 144$]],
  [Solve the equation.], [#math.equation(block: false, alt: "27 x equals 189")[$#h(3.7em) 27 x = 189$]],
  [], [#math.equation(block: false, alt: "x equals 7")[$#h(4.76em) x = 7$]],
  [Check the answer.], [],
  [#figure(figph[Verification of the equation 3 square root of (3x minus 5) minus 8 equals 4 using x equals 7. Substituting x equals 7 into the left side yields 3 square root of (3 times 7 minus 5) minus 8 equals 3 square root of (21 minus 5) minus 8 equals 3 square root of 16 minus 8 equals 3 times 4 minus 8 equals 12 minus 8 equals 4. The right side is 4, so 4 equals 4. This confirms that x equals 7 is a solution to the equation.], alt: "Verification of the equation 3 square root of (3x minus 5) minus 8 equals 4 using x equals 7. Substituting x equals 7 into the left side yields 3 square root of (3 times 7 minus 5) minus 8 equals 3 square root of (21 minus 5) minus 8 equals 3 square root of 16 minus 8 equals 3 times 4 minus 8 equals 12 minus 8 equals 4. The right side is 4, so 4 equals 4. This confirms that x equals 7 is a solution to the equation.", caption: none)], [The solution is #math.equation(block: false, alt: "x equals 7")[$x = 7$].],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "2 the square root of 4 a plus 2 minus 16 equals 16")[$2 sqrt(4 a + 2) − 16 = 16$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 127 over 2")[$frac(127, 2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "3 the square root of 6 b plus 3 minus 25 equals 50")[$3 sqrt(6 b + 3) − 25 = 50$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 311 over 3")[$frac(311, 3)$]
]
]

#examplebox("Example 9")[][
Solve: #math.equation(block: false, alt: "the square root of 4 z minus 3 equals the square root of 3 z plus 2")[$sqrt(4 z − 3) = sqrt(3 z + 2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 4 z minus 3 equals the square root of 3 z plus 2")[$#h(5em) sqrt(4 z − 3) = sqrt(3 z + 2)$]]),
  [The radical terms are isolated.], [#math.equation(block: false, alt: "the square root of 4 z minus 3 equals the square root of 3 z plus 2")[$#h(5em) sqrt(4 z − 3) = sqrt(3 z + 2)$]],
  [Square both sides of the equation.], [#math.equation(block: false, alt: "open parenthesis the square root of 4 z minus 3 close parenthesis squared equals open parenthesis the square root of 3 z plus 2 close parenthesis squared")[$#h(4.07em) attach(( sqrt(4 z − 3) ), t: 2) = attach(( sqrt(3 z + 2) ), t: 2)$]],
  [Simplify, then solve the new equation.], [#math.equation(block: false, alt: "4 z minus 3 equals 3 z plus 2; z minus 3 equals 2; z equals 5")[$#h(5.5em) 4 z − 3 = 3 z + 2 \ #h(6em) z − 3 = 2 \ #h(7.61em) z = 5$]],
  [Check the answer.], [],
  [We leave it to you to show that 5 checks!], [The solution is #math.equation(block: false, alt: "z equals 5 .")[$z = 5 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 2 x minus 5 equals the square root of 5 x plus 3")[$sqrt(2 x − 5) = sqrt(5 x + 3)$].

#solutionbox[
no solution
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of 7 y plus 1 equals the square root of 2 y minus 5")[$sqrt(7 y + 1) = sqrt(2 y − 5)$].

#solutionbox[
no solution
]
]

Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again.

#examplebox("Example 10")[][
Solve: #math.equation(block: false, alt: "the square root of m plus 1 equals the square root of m plus 9")[$sqrt(m) + 1 = sqrt(m + 9)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of m plus 1 equals the square root of m plus 9")[$#h(2.43em) sqrt(m) + 1 = sqrt(m + 9)$]]),
  [The radical on the right side is isolated. Square both sides.], [#math.equation(block: false, alt: "open parenthesis the square root of m plus 1 close parenthesis squared equals open parenthesis the square root of m plus 9 close parenthesis squared")[$#h(1.3em) attach(( sqrt(m) + 1 ), t: 2) = attach(( sqrt(m + 9) ), t: 2)$]],
  [Simplify—be very careful as you multiply!], [#math.equation(block: false, alt: "m plus 2 the square root of m plus 1 equals m plus 9")[$#h(0.09em) m + 2 sqrt(m) + 1 = m + 9$]],
  [There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical.], [#math.equation(block: false, alt: "2 the square root of m equals 8")[$#h(3.58em) 2 sqrt(m) = 8$]],
  [Square both sides.], [#math.equation(block: false, alt: "open parenthesis 2 the square root of m close parenthesis squared equals open parenthesis 8 close parenthesis squared")[$#h(2.43em) attach(( 2 sqrt(m) ), t: 2) = attach(( 8 ), t: 2)$]],
  [Simplify, then solve the new equation.], [#math.equation(block: false, alt: "4 m equals 64")[$#h(4.04em) 4 m = 64$]],
  [], [#math.equation(block: false, alt: "m equals 16")[$#h(4.54em) m = 16$]],
  [], [Check the answer.],
  [We leave it to you to show that #math.equation(block: false, alt: "m equals 16")[$m = 16$] checks!], [The solution is #math.equation(block: false, alt: "m equals 16 .")[$m = 16 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of x plus 3 equals the square root of x plus 5")[$sqrt(x) + 3 = sqrt(x + 5)$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of m plus 5 equals the square root of m plus 16")[$sqrt(m) + 5 = sqrt(m + 16)$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]

#examplebox("Example 11")[][
Solve: #math.equation(block: false, alt: "the square root of q minus 2 plus 3 equals the square root of 4 q plus 1")[$sqrt(q − 2) + 3 = sqrt(4 q + 1)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of q minus 2 plus 3 equals the square root of 4 q plus 1")[$#h(7.29em) sqrt(q − 2) + 3 = sqrt(4 q + 1)$]]),
  [The radical on the right side is isolated. Square both sides.], [#math.equation(block: false, alt: "open parenthesis the square root of q minus 2 plus 3 close parenthesis squared equals open parenthesis the square root of 4 q plus 1 close parenthesis squared")[$#h(6.34em) attach(( sqrt(q − 2) + 3 ), t: 2) = attach(( sqrt(4 q + 1) ), t: 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "q minus 2 plus 6 the square root of q minus 2 plus 9 equals 4 q plus 1")[$#h(3.57em) q − 2 + 6 sqrt(q − 2) + 9 = 4 q + 1$]],
  [There is still a radical in the equation. So we must repeat the previous steps. Isolate the radical.], [#math.equation(block: false, alt: "6 the square root of q minus 2 equals 3 q minus 6")[$#h(8.41em) 6 sqrt(q − 2) = 3 q − 6$]],
  [Square both sides.], [#math.equation(block: false, alt: "open parenthesis 6 the square root of q minus 2 close parenthesis squared equals open parenthesis 3 q minus 6 close parenthesis squared")[$#h(7.52em) attach(( 6 sqrt(q − 2) ), t: 2) = attach(( 3 q − 6 ), t: 2)$]],
  [Simplify, then solve the new equation.], [#math.equation(block: false, alt: "36 open parenthesis q minus 2 close parenthesis equals 9 q squared minus 36 q plus 36")[$#h(8.11em) 36 ( q − 2 ) = 9 q^(2) − 36 q + 36$]],
  [Distribute.], [#math.equation(block: false, alt: "36 q minus 72 equals 9 q squared minus 36 q plus 36")[$#h(7.88em) 36 q − 72 = 9 q^(2) − 36 q + 36$]],
  [It is a quadratic equation, so get zero on one side.], [#math.equation(block: false, alt: "0 equals 9 q squared minus 72 q plus 108")[$#h(11.06em) 0 = 9 q^(2) − 72 q + 108$]],
  [Factor the right side.], [#math.equation(block: false, alt: "0 equals 9 open parenthesis q squared minus 8 q plus 12 close parenthesis")[$#h(11.06em) 0 = 9 ( q^(2) − 8 q + 12 )$]],
  [], [#math.equation(block: false, alt: "0 equals 9 open parenthesis q minus 6 close parenthesis open parenthesis q minus 2 close parenthesis")[$#h(11.06em) 0 = 9 ( q − 6 ) ( q − 2 )$]],
  [Use the zero product property.], [#math.equation(block: false, alt: "q minus 6 equals 0, q minus 2 equals 0; q equals 6, q equals 2")[$#h(4.05em) q − 6 = 0 &  &  & q − 2 = 0 \ #h(5.65em) q = 6 &  &  & #h(1.65em) q = 2$]],
  [The checks are left to you. (Both solutions should work.)], [#math.equation(block: false, alt: "The solutions are q equals 6 and q equals 2 .")[$#h(4em) "The solutions are" #h(0.2em) q = 6 #h(0.2em) "and" #h(0.2em) q = 2 .$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of y minus 3 plus 2 equals the square root of 4 y plus 2")[$sqrt(y − 3) + 2 = sqrt(4 y + 2)$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Solve: #math.equation(block: false, alt: "the square root of n minus 4 plus 5 equals the square root of 3 n plus 3")[$sqrt(n − 4) + 5 = sqrt(3 n + 3)$].

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]
]


=== Use Square Roots in Applications

As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.

We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Solve applications with formulas.]

+ #strong[Read] the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #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 used the formula #math.equation(block: false, alt: "A equals L times W")[$A = L · W$] to find the area of a rectangle with length #emph[L] and width #emph[W]. A square is a rectangle in which the length and width are equal. If we let #emph[s] be the length of a side of a square, the area of the square is #math.equation(block: false, alt: "s squared")[$s^(2)$].

#figure(figph[This figure shows a square with two sides labeled s. It also indicates that A equals s squared.], alt: "This figure shows a square with two sides labeled s. It also indicates that A equals s squared.", caption: none)

The formula #math.equation(block: false, alt: "A equals s squared")[$A = s^(2)$] gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for #emph[s].

#math.equation(block: true, alt: "A equals s squared; Take the square root of both sides., the square root of A equals the square root of s squared; Simplify., the square root of A equals s")[$&  &  & #h(0.5em) A = s^(2) \ "Take the square root of both sides." &  &  & sqrt(A) = sqrt(s^(2)) \ "Simplify." &  &  & sqrt(A) = s$]

We can use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of a side of a square for a given area.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Area of a Square]

#figure(figph[This figure shows a square with two sides labeled s. The figure also indicates, “Area, A,” “A equals s squared,” “Length of a side, s,” and “s equals the square root of A.”], alt: "This figure shows a square with two sides labeled s. The figure also indicates, “Area, A,” “A equals s squared,” “Length of a side, s,” and “s equals the square root of A.”", caption: none)
]

We will show an example of this in the next example.

#examplebox("Example 12")[][
Mike and Lychelle want to make a square patio. They have enough concrete to pave an area of 200 square feet. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of the patio. Round your answer to the nearest tenth of a foot.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem. Draw a figure and #linebreak() label it with the given information.], [#figure(figph[A simple black outline of a square with the letter 's' labeling its left vertical side and its bottom horizontal side, indicating that all sides have a length of 's'.], alt: "A simple black outline of a square with the letter 's' labeling its left vertical side and its bottom horizontal side, indicating that all sides have a length of 's'.", caption: none)],
  [], [#emph[A] = 200 square feet],
  [#strong[Step 2. Identify] what you are looking for.], [The length of a side of the square patio.],
  [#strong[Step 3. Name] what you are looking for by #linebreak() choosing a variable to represent it.], [Let #emph[s] = the length of a side.],
  [#strong[Step 4. Translate] into an equation by writing the #linebreak() appropriate formula or model for the situation. #linebreak() Substitute the given information.], [#figure(figph[A mathematical equation illustrating the substitution of A = 200 into the formula s = sqrt(A), resulting in s = sqrt(200).], alt: "A mathematical equation illustrating the substitution of A = 200 into the formula s = sqrt(A), resulting in s = sqrt(200).", caption: none)],
  [#strong[Step 5. Solve the equation] using good algebra #linebreak() techniques. Round to one decimal place.], [#figure(figph[An example of rounding a decimal number: 's' equals 14.14213... is approximated as 14.1.], alt: "An example of rounding a decimal number: 's' equals 14.14213... is approximated as 14.1.", caption: none)],
  [#strong[Step 6. Check] the answer in the problem and #linebreak() make sure it makes sense.], [],
  [#figure(figph[A math problem asks to approximate 14.1 squared; it questions if 14.1^2 is approximately 200 and then provides the actual value: 14.1^2 = 198.81, with a checkmark.], alt: "A math problem asks to approximate 14.1 squared; it questions if 14.1^2 is approximately 200 and then provides the actual value: 14.1^2 = 198.81, with a checkmark.", caption: none)], [],
  [This is close enough because we rounded the #linebreak() square root. #linebreak() Is a patio with side 14.1 feet reasonable? #linebreak() Yes.], [],
  [#strong[Step 7. Answer] the question with a complete #linebreak() sentence.], [Each side of the patio should be 14.1 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of her lawn. Round your answer to the nearest tenth of a foot.

#solutionbox[
#math.equation(block: true, alt: "19.2 feet")[$19.2 #h(0.2em) "feet"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of his mosaic. Round your answer to the nearest tenth of a centimeter.

#solutionbox[
#math.equation(block: true, alt: "52.0 cm")[$52.0 #h(0.2em) "cm"$]
]
]

Another application of square roots has to do with gravity.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Falling Objects]

On Earth, if an object is dropped from a height of #math.equation(block: false, alt: "h")[$h$] feet, the time in seconds it will take to reach the ground is found by using the formula,

#math.equation(block: true, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$]
]

For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting #math.equation(block: false, alt: "h equals 64")[$h = 64$] into the formula.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The mathematical formula t = sqrt(h) / 4 is shown, representing a relationship between 't' and the square root of 'h' divided by 4.], alt: "The mathematical formula t = sqrt(h) / 4 is shown, representing a relationship between 't' and the square root of 'h' divided by 4.", caption: none)]),
  [], [#figure(figph[Equation showing t equals the square root of sixty-four divided by four. Sixty-four is highlighted.], alt: "Equation showing t equals the square root of sixty-four divided by four. Sixty-four is highlighted.", caption: none)],
  [Take the square root of 64.], [#figure(figph[A mathematical equation displays 't = 8/4' on a white background, representing the variable t being equal to the fraction eight divided by four.], alt: "A mathematical equation displays 't = 8/4' on a white background, representing the variable t being equal to the fraction eight divided by four.", caption: none)],
  [Simplify the fraction.], [#figure(figph[The text 't=2' is displayed in the center of a white background.], alt: "The text 't=2' is displayed in the center of a white background.", caption: none)],
))

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.

#examplebox("Example 13")[][
Christy dropped her sunglasses from a bridge 400 feet above a river. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the sunglasses to reach the river.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem.], [],
  [#strong[Step 2. Identify] what you are looking for.], [The time it takes for the sunglasses to reach #linebreak() the river.],
  [#strong[Step 3. Name] what you are looking for by #linebreak() choosing a variable to represent it.], [Let #emph[t] = time.],
  [#strong[Step 4. Translate] into an equation by writing the #linebreak() appropriate formula or model for the situation. #linebreak() Substitute in the given information.], [#figure(figph[The equations t equals square root of h divided by 4, and h equals 400.], alt: "The equations t equals square root of h divided by 4, and h equals 400.", caption: none)

 #linebreak() #figure(figph[A mathematical equation shows 't equals the square root of 400, all divided by 4.' The number 400 is highlighted in red, indicating it's a specific part of the calculation.], alt: "A mathematical equation shows 't equals the square root of 400, all divided by 4.' The number 400 is highlighted in red, indicating it's a specific part of the calculation.", caption: none)],
  [#strong[Step 5. Solve the equation] using good algebra #linebreak() techniques.], [#figure(figph[A mathematical equation is displayed on a white background, which reads 't = 20/4'.], alt: "A mathematical equation is displayed on a white background, which reads 't = 20/4'.", caption: none)

 #linebreak() #figure(figph[The image displays the mathematical expression 't = 5' in black characters against a plain white background.], alt: "The image displays the mathematical expression 't = 5' in black characters against a plain white background.", caption: none)],
  [#strong[Step 6. Check] the answer in the problem and #linebreak() make sure it makes sense.], [],
  [#figure(figph[A mathematical equation asks if 5 is equal to the square root of 400, divided by 4, represented as '5 ?= 400/4'. The question mark above the equals sign indicates verification.], alt: "A mathematical equation asks if 5 is equal to the square root of 400, divided by 4, represented as '5 ?= 400/4'. The question mark above the equals sign indicates verification.", caption: none)

 #linebreak() #figure(figph[A math equation displays 5 followed by a question mark over an equals sign, then the fraction 20 over 4, asking if 5 is equal to 20 divided by 4.], alt: "A math equation displays 5 followed by a question mark over an equals sign, then the fraction 20 over 4, asking if 5 is equal to 20 divided by 4.", caption: none)

 #linebreak() #math.equation(block: false, alt: "5 equals 5 ✓")[$5 = 5 ✓ #h(4.5em)$]], [],
  [Does 5 seconds seem reasonable? #linebreak() Yes.], [],
  [#strong[Step 7. Answer] the question with a complete #linebreak() sentence.], [It will take 5 seconds for the sunglasses to hit #linebreak() the water.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the package to reach the ground.

#solutionbox[
#math.equation(block: true, alt: "9 seconds")[$9 #h(0.2em) "seconds"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the squeegee to reach the sidewalk.

#solutionbox[
#math.equation(block: true, alt: "3.5 seconds")[$3.5 #h(0.2em) "seconds"$]
]
]

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Skid Marks and Speed of a Car]

If the length of the skid marks is #emph[d] feet, then the speed, #emph[s], of the car before the brakes were applied can be found by using the formula,

#math.equation(block: true, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$]
]

#examplebox("Example 14")[][
After a car accident, the skid marks for one car measured 190 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Step 1. #strong[Read] the problem.], []),
  [Step 2. #strong[Identify] what we are looking for.], [The speed of a car.],
  [Step 3. #strong[Name] what we are looking for.], [Let s = the speed.],
  [Step 4. #strong[Translate] into an equation by writing the appropriate formula.], [#figure(figph[A mathematical problem showing the equation s = sqrt(24d) and the value d = 190, likely for solving 's'.], alt: "A mathematical problem showing the equation s = sqrt(24d) and the value d = 190, likely for solving 's'.", caption: none)],
  [Substitute the given information.], [#figure(figph[A mathematical equation displays 's' equals the square root of 24 times 190, with the number 190 highlighted in red, indicating a specific focus or variable.], alt: "A mathematical equation displays 's' equals the square root of 24 times 190, with the number 190 highlighted in red, indicating a specific focus or variable.", caption: none)],
  [Step 5. #strong[Solve the equation].], [#figure(figph[A mathematical equation shows 's = ', followed by a square root symbol over the number '4560'.], alt: "A mathematical equation shows 's = ', followed by a square root symbol over the number '4560'.", caption: none)],
  [], [#figure(figph[The image displays a mathematical equation or variable assignment, 's = 67.52777...', where 's' is assigned a repeating decimal value. The text is clear and centrally positioned on a white background.], alt: "The image displays a mathematical equation or variable assignment, 's = 67.52777...', where 's' is assigned a repeating decimal value. The text is clear and centrally positioned on a white background.", caption: none)],
  [Round to 1 decimal place.], [#figure(figph[A mathematical expression, 's is approximately 67.5'.], alt: "A mathematical expression, 's is approximately 67.5'.", caption: none)],
  [Step 6. #strong[Check] the answer in the problem. #linebreak() #math.equation(block: false, alt: "67.5 approximately equals ? the square root of 24 open parenthesis 190 close parenthesis")[$#h(1em) 67.5 limits(≈)^(?) sqrt(24 ( 190 ))$] #linebreak() #math.equation(block: false, alt: "67.5 approximately equals ? the square root of 4560")[$#h(1em) 67.5 limits(≈)^(?) #h(0.2em) sqrt(4560)$] #linebreak() #math.equation(block: false, alt: "67.5 approximately equals ? 67.5277...")[$#h(1em) 67.5 limits(≈)^(?) #h(0.2em) 67.5277...$]], [],
  [Is 67.5 mph a reasonable speed?], [Yes.],
  [Step 7. #strong[Answer] the question with a complete sentence.], [The speed of the car was approximately 67.5 miles per hour.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: true, alt: "42.7 miles per hour")[$42.7 #h(0.2em) "miles per hour"$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: true, alt: "54.1 miles per hour")[$54.1 #h(0.2em) "miles per hour"$]
]
]


=== Key Concepts

- #strong[To Solve a Radical Equation:]
+ Isolate the radical on one side of the equation.
+ Square both sides of the equation.
+ Solve the new equation.
+ Check the answer. Some solutions obtained may not work in the original equation.
- #strong[Solving Applications with Formulas]
+ #strong[Read] the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
+ #strong[Identify] what we are looking for.
+ #strong[Name] what we are looking for by choosing a variable to represent it.
+ #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
+ #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.
- #strong[Area of a Square] #linebreak() 
#figure(figph[This figure shows a square with two sides labeled, “s.” The figure also says, “Area, A,” “A equals s squared,” “Length of a side, s,” and “s equals the square root of A.”], alt: "This figure shows a square with two sides labeled, “s.” The figure also says, “Area, A,” “A equals s squared,” “Length of a side, s,” and “s equals the square root of A.”", caption: none)
- #strong[Falling Objects]
- On Earth, if an object is dropped from a height of #math.equation(block: false, alt: "h")[$h$] feet, the time in seconds it will take to reach the ground is found by using the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$].
- #strong[Skid Marks and Speed of a Car]
- If the length of the skid marks is #emph[d] feet, then the speed, #emph[s], of the car before the brakes were applied can be found by using the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$].


==== Practice Makes Perfect

#strong[Solve Radical Equations]

In the following exercises, check whether the given values are solutions.

For the equation #math.equation(block: false, alt: "the square root of x plus 12 equals x")[$sqrt(x + 12) = x$]:
ⓐ Is #math.equation(block: false, alt: "x equals 4")[$x = 4$] a solution? ⓑ Is #math.equation(block: false, alt: "x equals −3")[$x = −3$] a solution?

#solutionbox[
ⓐ #math.equation(block: false, alt: "yes")[$"yes"$] ⓑ #math.equation(block: false, alt: "no")[$"no"$]
]

For the equation #math.equation(block: false, alt: "the square root of − y plus 20 equals y")[$sqrt("−" y + 20) = y$]:
ⓐ Is #math.equation(block: false, alt: "y equals 4")[$y = 4$] a solution? ⓑ Is #math.equation(block: false, alt: "y equals −5")[$y = −5$] a solution?

For the equation #math.equation(block: false, alt: "the square root of t plus 6 equals t")[$sqrt(t + 6) = t$]:
ⓐ Is #math.equation(block: false, alt: "t equals −2")[$t = −2$] a solution? ⓑ Is #math.equation(block: false, alt: "t equals 3")[$t = 3$] a solution?

#solutionbox[
ⓐ no ⓑ yes
]

For the equation #math.equation(block: false, alt: "the square root of u plus 42 equals u")[$sqrt(u + 42) = u$]:
ⓐ Is #math.equation(block: false, alt: "u equals −6")[$u = −6$] a solution? ⓑ Is #math.equation(block: false, alt: "u equals 7")[$u = 7$] a solution?

In the following exercises, solve.

#math.equation(block: true, alt: "the square root of 5 y plus 1 equals 4")[$sqrt(5 y + 1) = 4$]

#solutionbox[
3
]

#math.equation(block: true, alt: "the square root of 7 z plus 15 equals 6")[$sqrt(7 z + 15) = 6$]

#math.equation(block: true, alt: "the square root of 5 x minus 6 equals 8")[$sqrt(5 x − 6) = 8$]

#solutionbox[
14
]

#math.equation(block: true, alt: "the square root of 4 x minus 3 equals 7")[$sqrt(4 x − 3) = 7$]

#math.equation(block: true, alt: "the square root of 2 m minus 3 minus 5 equals 0")[$sqrt(2 m − 3) − 5 = 0$]

#solutionbox[
14
]

#math.equation(block: true, alt: "the square root of 2 n minus 1 minus 3 equals 0")[$sqrt(2 n − 1) − 3 = 0$]

#math.equation(block: true, alt: "the square root of 6 v minus 2 minus 10 equals 0")[$sqrt(6 v − 2) − 10 = 0$]

#solutionbox[
#math.equation(block: true, alt: "17")[$17$]
]

#math.equation(block: true, alt: "the square root of 4 u plus 2 minus 6 equals 0")[$sqrt(4 u + 2) − 6 = 0$]

#math.equation(block: true, alt: "the square root of 5 q plus 3 minus 4 equals 0")[$sqrt(5 q + 3) − 4 = 0$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 13 over 5")[$frac(13, 5)$]
]

#math.equation(block: true, alt: "the square root of 4 m plus 2 plus 2 equals 6")[$sqrt(4 m + 2) + 2 = 6$]

#math.equation(block: true, alt: "the square root of 6 n plus 1 plus 4 equals 8")[$sqrt(6 n + 1) + 4 = 8$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 5 over 2")[$frac(5, 2)$]
]

#math.equation(block: true, alt: "the square root of 2 u minus 3 plus 2 equals 0")[$sqrt(2 u − 3) + 2 = 0$]

#math.equation(block: true, alt: "the square root of 5 v minus 2 plus 5 equals 0")[$sqrt(5 v − 2) + 5 = 0$]

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]

#math.equation(block: true, alt: "the square root of 3 z minus 5 plus 2 equals 0")[$sqrt(3 z − 5) + 2 = 0$]

#math.equation(block: true, alt: "the square root of 2 m plus 1 plus 4 equals 0")[$sqrt(2 m + 1) + 4 = 0$]

#solutionbox[
#math.equation(block: true, alt: "no solution")[$"no solution"$]
]

ⓐ #math.equation(block: false, alt: "the square root of u minus 3 plus 3 equals u")[$sqrt(u − 3) + 3 = u$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of x plus 1 minus x plus 1 equals 0")[$sqrt(x + 1) − x + 1 = 0$]

ⓐ #math.equation(block: false, alt: "the square root of v minus 10 plus 10 equals v")[$sqrt(v − 10) + 10 = v$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of y plus 4 minus y plus 2 equals 0")[$sqrt(y + 4) − y + 2 = 0$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "10 , 11")[$10 , 11$] ⓑ #math.equation(block: false, alt: "5")[$5$]
]

ⓐ #math.equation(block: false, alt: "the square root of r minus 1 minus r equals −1")[$sqrt(r − 1) − r = −1$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of z plus 100 minus z plus 10 equals 0")[$sqrt(z + 100) − z + 10 = 0$]

ⓐ #math.equation(block: false, alt: "the square root of s minus 8 minus s equals −8")[$sqrt(s − 8) − s = −8$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of w plus 25 minus w plus 5 equals 0")[$sqrt(w + 25) − w + 5 = 0$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "8 , 9")[$8 , 9$] ⓑ #math.equation(block: false, alt: "11")[$11$]
]

#math.equation(block: true, alt: "3 the square root of 2 x minus 3 minus 20 equals 7")[$3 sqrt(2 x − 3) − 20 = 7$]

#math.equation(block: true, alt: "2 the square root of 5 x plus 1 minus 8 equals 0")[$2 sqrt(5 x + 1) − 8 = 0$]

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

#math.equation(block: true, alt: "2 the square root of 8 r plus 1 minus 8 equals 2")[$2 sqrt(8 r + 1) − 8 = 2$]

#math.equation(block: true, alt: "3 the square root of 7 y plus 1 minus 10 equals 8")[$3 sqrt(7 y + 1) − 10 = 8$]

#solutionbox[
#math.equation(block: true, alt: "5")[$5$]
]

#math.equation(block: true, alt: "the square root of 3 u minus 2 equals the square root of 5 u plus 1")[$sqrt(3 u − 2) = sqrt(5 u + 1)$]

#math.equation(block: true, alt: "the square root of 4 v plus 3 equals the square root of v minus 6")[$sqrt(4 v + 3) = sqrt(v − 6)$]

#solutionbox[
not a real number
]

#math.equation(block: true, alt: "the square root of 8 plus 2 r equals the square root of 3 r plus 10")[$sqrt(8 + 2 r) = sqrt(3 r + 10)$]

#math.equation(block: true, alt: "the square root of 12 c plus 6 equals the square root of 10 minus 4 c")[$sqrt(12 c + 6) = sqrt(10 − 4 c)$]

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

ⓐ #math.equation(block: false, alt: "the square root of a plus 2 equals the square root of a plus 4")[$sqrt(a) + 2 = sqrt(a + 4)$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of b minus 2 plus 1 equals the square root of 3 b plus 2")[$sqrt(b − 2) + 1 = sqrt(3 b + 2)$]

ⓐ #math.equation(block: false, alt: "the square root of r plus 6 equals the square root of r plus 8")[$sqrt(r) + 6 = sqrt(r + 8)$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of s minus 3 plus 2 equals the square root of s plus 4")[$sqrt(s − 3) + 2 = sqrt(s + 4)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "no solution")[$"no solution"$] ⓑ #math.equation(block: false, alt: "the fraction 57 over 16")[$frac(57, 16)$]
]

ⓐ #math.equation(block: false, alt: "the square root of u plus 1 equals the square root of u plus 4")[$sqrt(u) + 1 = sqrt(u + 4)$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of n minus 5 plus 4 equals the square root of 3 n plus 7")[$sqrt(n − 5) + 4 = sqrt(3 n + 7)$]

ⓐ #math.equation(block: false, alt: "the square root of x plus 10 equals the square root of x plus 2")[$sqrt(x) + 10 = sqrt(x + 2)$] #linebreak() ⓑ #math.equation(block: false, alt: "the square root of y minus 2 plus 2 equals the square root of 2 y plus 4")[$sqrt(y − 2) + 2 = sqrt(2 y + 4)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "no solution")[$"no solution"$] ⓑ 6
]

#math.equation(block: true, alt: "the square root of 2 y plus 4 plus 6 equals 0")[$sqrt(2 y + 4) + 6 = 0$]

#math.equation(block: true, alt: "the square root of 8 u plus 1 plus 9 equals 0")[$sqrt(8 u + 1) + 9 = 0$]

#solutionbox[
no solution
]

#math.equation(block: true, alt: "the square root of a plus 1 equals the square root of a plus 5")[$sqrt(a) + 1 = sqrt(a + 5)$]

#math.equation(block: true, alt: "the square root of d minus 2 equals the square root of d minus 20")[$sqrt(d) − 2 = sqrt(d − 20)$]

#solutionbox[
36
]

#math.equation(block: true, alt: "the square root of 6 s plus 4 equals the square root of 8 s minus 28")[$sqrt(6 s + 4) = sqrt(8 s − 28)$]

#math.equation(block: true, alt: "the square root of 9 p plus 9 equals the square root of 10 p minus 6")[$sqrt(9 p + 9) = sqrt(10 p − 6)$]

#solutionbox[
15
]

#strong[Use Square Roots in Applications]

In the following exercises, solve. Round approximations to one decimal place.

#strong[Landscaping] Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of 75 square feet. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.

#strong[Landscaping] Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

#solutionbox[
#math.equation(block: true, alt: "11.4 feet")[$11.4 #h(0.2em) "feet"$]
]

#strong[Gravity] While putting up holiday decorations, Renee dropped a light bulb from the top of a 64 foot tall tree. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the light bulb to reach the ground.

#strong[Gravity] An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the flare to reach the water.

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

#strong[Gravity] A hang glider dropped his cell phone from a height of 350 feet. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the cell phone to reach the ground.

#strong[Gravity] A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the hammer to reach the river.

#solutionbox[
#math.equation(block: true, alt: "15.8 seconds")[$15.8 #h(0.2em) "seconds"$]
]

#strong[Accident investigation] The skid marks for a car involved in an accident measured 54 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

#strong[Accident investigation] The skid marks for a car involved in an accident measured 216 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: true, alt: "72 miles per hour")[$72 #h(0.2em) "miles per hour"$]
]

#strong[Accident investigation] An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 175 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

#strong[Accident investigation] An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: true, alt: "53.0 miles per hour")[$53.0 #h(0.2em) "miles per hour"$]
]


==== Writing Exercises

Explain why an equation of the form #math.equation(block: false, alt: "the square root of x plus 1 equals 0")[$sqrt(x) + 1 = 0$] has no solution.

+ ⓐ Solve the equation #math.equation(block: false, alt: "the square root of r plus 4 minus r plus 2 equals 0")[$sqrt(r + 4) − r + 2 = 0$].
+ ⓑ Explain why one of the “solutions” that was found was not actually a solution to the equation.

#solutionbox[
Answers will vary.
]


==== Self Check

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

#figure(figph[This table has two rows and four columns. The first row labels each column, “I can…,” “Confidently,” “With some help,” and “No minus I don’t get it!” The row under “I can…,” reads, “use square roots in applications.” All the other rows are empty.], alt: "This table has two rows and four columns. The first row labels each column, “I can…,” “Confidently,” “With some help,” and “No minus I don’t get it!” The row under “I can…,” reads, “use square roots in applications.” All the other rows are empty.", caption: none)

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?