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== 10.4#h(0.6em)Solve Applications Modeled by Quadratic Equations

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

The sum of two consecutive odd numbers is #math.equation(block: false, alt: "−100")[$−100$]. Find the numbers. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "minus 51 , minus 49")[$− 51 , − 49$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The area of triangular mural is 64 square feet. The base is 16 feet. Find the height. #linebreak() 
If you missed this problem, review .

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the length of the hypotenuse of a right triangle with legs 5 inches and 12 inches. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "13 inches")[$13 " " "inches"$]
]
]


=== Solve Applications of the Quadratic Formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Use the problem solving strategy.]

+ #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 have solved number applications that involved #strong[consecutive even integers] and #strong[consecutive odd integers] by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is #math.equation(block: false, alt: "n plus 2")[$n + 2$]. The next one would be #math.equation(block: false, alt: "n plus 2 plus 2")[$n + 2 + 2$] or #math.equation(block: false, alt: "n plus 4")[$n + 4$]. This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.

#math.equation(block: true, alt: "Consecutive even integers, Consecutive odd integers; 64 , 66 , 68, 77 , 79 , 81; n, 1 to the power st even integer; n plus 2, 2 to the power nd consecutive even integer; n plus 4, 3 to the power rd consecutive even integer, n, 1 to the power st odd integer; n plus 2, 2 to the power nd consecutive odd integer; n plus 4, 3 to the power rd consecutive odd integer")[$"Consecutive even integers" &  &  & #h(4em) "Consecutive odd integers" \ 64 , 66 , 68 &  &  & #h(4em) 77 , 79 , 81 \ n &  &  & 1^("st") #h(0.2em) "even integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "consecutive even integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive even integer" &  &  & #h(4em) n &  &  & 1^("st") #h(0.2em) "odd integer" \ n + 2 &  &  & 2^("nd") #h(0.2em) "consecutive odd integer" \ n + 4 &  &  & 3^("rd") #h(0.2em) "consecutive odd integer"$]Some applications of #strong[consecutive odd integers] or #strong[consecutive even integers] are modeled by quadratic equations. The notation above will be helpful as you name the variables.

#examplebox("Example 1")[][
The product of two consecutive odd integers is 195. Find the integers.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1. Read] the problem.], []),
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for two consecutive odd integers.],
  [#strong[Step 3. Name] what we are looking for.], [Let #math.equation(block: false, alt: "n equals")[$n =$] the first odd integer. #linebreak() #math.equation(block: false, alt: "n plus 2 equals")[$n + 2 =$] the next odd integer],
  [#strong[Step 4. Translate] into an equation. State the problem in one sentence.], ["The product of two consecutive odd integers is 195." The product of the first odd integer and the second odd integer is 195.],
  [Translate into an equation], [#figure(figph[The image displays the mathematical equation n(n+2) = 195, where 'n' is a variable.], alt: "The image displays the mathematical equation n(n+2) = 195, where 'n' is a variable.", caption: none)],
  [#strong[Step 5. Solve] the equation. Distribute.], [#figure(figph[A mathematical equation is displayed on a white background, reading 'n squared plus 2n equals 195'.], alt: "A mathematical equation is displayed on a white background, reading 'n squared plus 2n equals 195'.", caption: none)],
  [Subtract 195 to get the equation in standard form.], [#figure(figph[Two quadratic equations are displayed: the general form 'ax^2 + bx + c = 0' in red, followed by a specific example, 'n^2 + 2n - 195 = 0,' in black text.], alt: "Two quadratic equations are displayed: the general form 'ax^2 + bx + c = 0' in red, followed by a specific example, 'n^2 + 2n - 195 = 0,' in black text.", caption: none)],
  [Identify the #emph[a, b, c] values.], [#figure(figph[The image displays mathematical variables with their assigned values: a = 1, b = 2, and c = -195, presented in different colors against a white background.], alt: "The image displays mathematical variables with their assigned values: a = 1, b = 2, and c = -195, presented in different colors against a white background.", caption: none)],
  [Write the quadratic equation.], [#figure(figph[The quadratic formula, solving for 'n' using coefficients a, b, and c.], alt: "The quadratic formula, solving for 'n' using coefficients a, b, and c.", caption: none)],
  [Then substitute in the values of #emph[a, b, c.].], [#figure(figph[The quadratic formula is shown with specific values substituted for solving n. The equation features -2 ×1 square root of (2^2 - 4 \* 1 \* -195), all divided by (2 \* 1), with some values highlighted.], alt: "The quadratic formula is shown with specific values substituted for solving n. The equation features -2 ×1 square root of (2^2 - 4 * 1 * -195), all divided by (2 * 1), with some values highlighted.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation is displayed, showing n equals a fraction where the numerator is -2 plus or minus the square root of 4 plus 780, all divided by 2.], alt: "A mathematical equation is displayed, showing n equals a fraction where the numerator is -2 plus or minus the square root of 4 plus 780, all divided by 2.", caption: none)

 #linebreak() #figure(figph[A mathematical equation shows 'n equals negative 2 plus or minus the square root of 784, all divided by 2' on a white background. This is a step in solving a quadratic equation.], alt: "A mathematical equation shows 'n equals negative 2 plus or minus the square root of 784, all divided by 2' on a white background. This is a step in solving a quadratic equation.", caption: none)],
  [Simplify the radical.], [#figure(figph[A mathematical equation shows 'n' is equal to a fraction where the numerator is '-2 plus or minus 28' and the denominator is '2'.], alt: "A mathematical equation shows 'n' is equal to a fraction where the numerator is '-2 plus or minus 28' and the denominator is '2'.", caption: none)],
  [Rewrite to show two solutions.], [#figure(figph[The image displays two expressions for 'n': n = (-2 + 28) / 2 and n = (-2 - 28) / 2. These typically represent the two roots found when solving a quadratic equation.], alt: "The image displays two expressions for 'n': n = (-2 + 28) / 2 and n = (-2 - 28) / 2. These typically represent the two roots found when solving a quadratic equation.", caption: none)],
  [Solve each equation.], [#figure(figph[Two mathematical equations are displayed horizontally on a white background: 'n = 26/2' and 'n = -30/2'. The equations show the variable 'n' being defined by fractions.], alt: "Two mathematical equations are displayed horizontally on a white background: 'n = 26/2' and 'n = -30/2'. The equations show the variable 'n' being defined by fractions.", caption: none)

 #linebreak() #figure(figph[The image displays the equations 'n = 13' and 'n = -15' in a simple, clear font on a white background.], alt: "The image displays the equations 'n = 13' and 'n = -15' in a simple, clear font on a white background.", caption: none)],
  [There are two values of #emph[n] that are solutions. This will give us two pairs of consecutive odd integers for our solution.], [First odd integer #math.equation(block: false, alt: "n equals 13")[$n = 13$] #linebreak() next odd integer #math.equation(block: false, alt: "n plus 2")[$n + 2$] #linebreak() #math.equation(block: false, alt: "13 plus 2")[$#h(6.8em) 13 + 2$] #linebreak() #math.equation(block: false, alt: "15")[$#h(8em) 15$]],
  [], [First odd integer #math.equation(block: false, alt: "n equals −15")[$n = −15$] #linebreak() next odd integer #math.equation(block: false, alt: "n plus 2")[$n + 2$] #linebreak() #math.equation(block: false, alt: "−15 plus 2")[$#h(6.1em) −15 + 2$] #linebreak() #math.equation(block: false, alt: "−13")[$#h(7em) −13$]],
  [#strong[Step 6. Check] the answer. #linebreak() Do these pairs work? #linebreak() Are they consecutive odd integers? #linebreak() Is their product 195?], [#linebreak()  #linebreak() #math.equation(block: false, alt: "13 , 15 , yes, −13 , −15 , yes; 13 times 15 equals 195 , yes, −13 open parenthesis −15 close parenthesis equals 195 , yes")[$13 , 15 , #h(0.2em) "yes" &  &  & −13 , −15 , #h(0.2em) "yes" \ 13 ⋅ 15 = 195 , #h(0.2em) "yes" &  &  & −13 ( −15 ) = 195 , #h(0.2em) "yes"$]],
  [#strong[Step 7. Answer] the question.], [The two consecutive odd integers whose product is 195 are 13, 15, and −13, −15.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The product of two consecutive odd integers is 99. Find the integers.

#solutionbox[
Two consecutive odd numbers whose product is 99 are 9 and 11, and #math.equation(block: false, alt: "−9")[$−9$] and #math.equation(block: false, alt: "−11")[$−11$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The product of two consecutive even integers is 168. Find the integers.

#solutionbox[
Two consecutive even numbers whose product is 168 are 12 and 14,and #math.equation(block: false, alt: "−12")[$−12$] and #math.equation(block: false, alt: "−14")[$−14$].
]
]

We will use the formula for the area of a triangle to solve the next example.

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

For a triangle with base #math.equation(block: false, alt: "b")[$b$] and height #math.equation(block: false, alt: "h")[$h$], the area, #math.equation(block: false, alt: "A")[$A$], is given by the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$].

#figure(figph[The image shows a triangle with a horizontal side at the bottom labeled b and a vertical line coming up from the side b to the vertex of the other two sides of the triangle. This vertical line is labeled h.], alt: "The image shows a triangle with a horizontal side at the bottom labeled b and a vertical line coming up from the side b to the vertex of the other two sides of the triangle. This vertical line is labeled h.", caption: none)
]

Recall that, when we solve geometry applications, it is helpful to draw the figure.

#examplebox("Example 2")[][
An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can have an area of 120 square feet and the architect wants the width to be 4 feet more than twice the height. Find the height and width of the window.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1. Read] the problem. #linebreak() Draw a picture.], [#figure(figph[A triangle with its height labeled 'h' and its base labeled '2h + 4', indicating dimensions for a geometric problem or calculation.], alt: "A triangle with its height labeled 'h' and its base labeled '2h + 4', indicating dimensions for a geometric problem or calculation.", caption: none)]),
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for the height and width.],
  [#strong[Step 3. Name] what we are looking for.], [Let #math.equation(block: false, alt: "h equals")[$h =$] the height of the triangle. #linebreak() #math.equation(block: false, alt: "2 h plus 4 equals")[$2 h + 4 =$] the width of the triangle],
  [#strong[Step 4. Translate.]], [We know the area. Write the formula for the area of a triangle.  #linebreak() #figure(figph[The image displays the mathematical formula for the area of a triangle, A = (1/2)bh, where A represents the area, b is the base, and h is the height.], alt: "The image displays the mathematical formula for the area of a triangle, A = (1/2)bh, where A represents the area, b is the base, and h is the height.", caption: none)],
  [#strong[Step 5. Solve] the equation. Substitute in the values.], [#figure(figph[The image shows the mathematical equation '120 = 1/2(2h + 4)h' centered on a white background. The equation involves numbers and the variable 'h', representing a quadratic relationship.], alt: "The image shows the mathematical equation '120 = 1/2(2h + 4)h' centered on a white background. The equation involves numbers and the variable 'h', representing a quadratic relationship.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation 120 = h² + 2h is displayed in the center of a white background.], alt: "A mathematical equation 120 = h² + 2h is displayed in the center of a white background.", caption: none)],
  [This is a quadratic equation, rewrite it in standard form.], [#figure(figph[Two quadratic equations are presented: the general form ax^2 + bx + c = 0 in red, and a specific instance h^2 + 2h - 120 = 0 in black on a white background.], alt: "Two quadratic equations are presented: the general form ax^2 + bx + c = 0 in red, and a specific instance h^2 + 2h - 120 = 0 in black on a white background.", caption: none)],
  [Solve the equation using the Quadratic Formula. Identify the #emph[a, b, c] values.], [#figure(figph[A mathematical expression displaying variables and their values: a = 1 (light blue), b = 2 (red), and c = -√120 (yellow-green), all on a white background.], alt: "A mathematical expression displaying variables and their values: a = 1 (light blue), b = 2 (red), and c = -√120 (yellow-green), all on a white background.", caption: none)],
  [Write the quadratic equation.], [#figure(figph[The quadratic formula is displayed, showing h equals the quantity negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.], alt: "The quadratic formula is displayed, showing h equals the quantity negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.", caption: none)],
  [Then substitute in the values of #emph[a, b, c.].], [#figure(figph[Mathematical expression showing the quadratic formula applied to find 'h', with specific values substituted.], alt: "Mathematical expression showing the quadratic formula applied to find 'h', with specific values substituted.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical equation for 'h' is shown: h equals a fraction where the numerator is -2 plus or minus the square root of (4 plus 480), and the denominator is 2.], alt: "A mathematical equation for 'h' is shown: h equals a fraction where the numerator is -2 plus or minus the square root of (4 plus 480), and the denominator is 2.", caption: none)

 #linebreak() #figure(figph[The image shows the mathematical equation h = (-2 ± sqrt(484)) / 2.], alt: "The image shows the mathematical equation h = (-2 ± sqrt(484)) / 2.", caption: none)],
  [Simplify the radical.], [#figure(figph[A mathematical equation is displayed on a white background: h = (-2  A mathematical equation is displayed on a white background: h = (-2 ± 22) / 2.], alt: "A mathematical equation is displayed on a white background: h = (-2 A mathematical equation is displayed on a white background: h = (-2 ± 22) / 2.", caption: none)],
  [Rewrite to show two solutions.], [#figure(figph[The two values for 'h' from a quadratic solution: h = (-2 + 22) / 2 and h = (-2 - 22) / 2.], alt: "The two values for 'h' from a quadratic solution: h = (-2 + 22) / 2 and h = (-2 - 22) / 2.", caption: none)],
  [Simplify.], [#figure(figph[Two mathematical equations are displayed on a white background. The first equation is h = 20/2, and the second is h = -24/2, with a comma separating the two expressions.], alt: "Two mathematical equations are displayed on a white background. The first equation is h = 20/2, and the second is h = -24/2, with a comma separating the two expressions.", caption: none)],
  [Since #emph[h] is the height of a window, a value of #math.equation(block: false, alt: "h equals − 12")[$h = "−" 12$] does not make sense.], [#figure(figph[Mathematical notation featuring 'h = 10' and the full expression 'b = 12' crossed out, suggesting a correction or an abandoned calculation.], alt: "Mathematical notation featuring 'h = 10' and the full expression 'b = 12' crossed out, suggesting a correction or an abandoned calculation.", caption: none)],
  [], [The height of the triangle: #math.equation(block: false, alt: "h equals 10")[$#h(1em) h = 10$] #linebreak()  #linebreak() The width of the triangle: #math.equation(block: false, alt: "2 h plus 4")[$#h(1.5em) 2 h + 4$] #linebreak() #math.equation(block: false, alt: "2 times 10 plus 4")[$#h(11.8em) 2 ⋅ 10 + 4$] #linebreak() #math.equation(block: false, alt: "24")[$#h(13.2em) 24$]],
  [#strong[Step 6. Check] the answer. Does a triangle with a height 10 and width 24 have area 120? Yes.], [],
  [#strong[Step 7. Answer] the question.], [The height of the triangular window is 10 feet and the width is 24 feet.],
))

Notice that the solutions were integers. That tells us that we could have solved the equation by factoring.

When we wrote the equation in standard form, #math.equation(block: false, alt: "h squared plus 2 h minus 120 equals 0")[$h^(2) + 2 h − 120 = 0$], we could have factored it. If we did, we would have solved the equation #math.equation(block: false, alt: "open parenthesis h plus 12 close parenthesis open parenthesis h minus 10 close parenthesis equals 0")[$( h + 12 ) ( h − 10 ) = 0$].
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Find the dimensions of a triangle whose width is four more than six times its height and has an area of 208 square inches.

#solutionbox[
The height of the triangle is 8 inches and the width is 52 inches.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
If a triangle that has an area of 110 square feet has a height that is two feet less than twice the width, what are its dimensions?

#solutionbox[
The height of the triangle is 20 feet and the width is 11 feet.
]
]

In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.

The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Pythagorean Theorem]

In any right triangle, where #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are the lengths of the legs and #math.equation(block: false, alt: "c")[$c$] is the length of the hypotenuse, #math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$]

#figure(figph[The image shows a right triangle with a horizontal side at the bottom labeled b, a vertical side on the left labeled a and the hypotenuse connecting the two is labeled c.], alt: "The image shows a right triangle with a horizontal side at the bottom labeled b, a vertical side on the left labeled a and the hypotenuse connecting the two is labeled c.", caption: none)
]

#examplebox("Example 3")[][
Rene is setting up a holiday light display. He wants to make a ‘tree’ in the shape of two right triangles, as shown below, and has two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be?

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1. Read] the problem. Draw a picture.], [#figure(figph[A geometric diagram displaying a triangle with a vertical line segment from the apex to the base. One of the sloped sides is labeled with the number '10,' and two downward arrows indicate points at the base.], alt: "A geometric diagram displaying a triangle with a vertical line segment from the apex to the base. One of the sloped sides is labeled with the number '10,' and two downward arrows indicate points at the base.", caption: none)]),
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for the height of the pole.],
  [#strong[Step 3. Name] what we are looking for.], [The distance from the base of the pole to either stake is the same as the height of the pole. Let #math.equation(block: false, alt: "x equals")[$x =$] the height of the pole.  #linebreak() #math.equation(block: false, alt: "x equals")[$x =$] the distance from the pole to stake],
  [Each side is a right triangle. We draw a picture of one of them.], [#figure(figph[A right-angled triangle is shown with two equal sides labeled 'x' and the hypotenuse labeled '10'.], alt: "A right-angled triangle is shown with two equal sides labeled 'x' and the hypotenuse labeled '10'.", caption: none)],
  [#strong[Step 4. Translate] into an equation. We can use the Pythagorean Theorem to solve for #emph[x].], [],
  [Write the Pythagorean Theorem.], [#math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$]],
  [#strong[Step 5. Solve] the equation. Substitute.], [#math.equation(block: false, alt: "x squared plus x squared equals 10 squared")[$x^(2) + x^(2) = 10^(2)$]],
  [Simplify.], [#math.equation(block: false, alt: "2 x squared equals 100")[$#h(1.6em) 2 x^(2) = 100$]],
  [Divide by 2 to isolate the variable.], [#math.equation(block: false, alt: "the fraction 2 x squared over 2 equals the fraction 100 over 2")[$#h(1.5em) frac(2 x^(2), 2) = frac(100, 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared equals 50")[$#h(2.1em) x^(2) = 50$]],
  [Use the Square Root Property.], [#math.equation(block: false, alt: "x equals plus or minus the square root of 50")[$#h(2.6em) x = ± #h(0.2em) sqrt(50)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "x equals plus or minus 5 the square root of 2")[$#h(2.6em) x = ± #h(0.2em) 5 sqrt(2)$]],
  [Rewrite to show two solutions.], [#math.equation(block: false, alt: "x equals 5 the square root of 2")[$#h(2.6em) x = 5 sqrt(2)$] #linebreak() #math.equation(block: false, alt: "x equals − 5 the square root of 2")[$#h(2.6em) cancel(x = "−" 5 sqrt(2))$]],
  [Approximate this number to the nearest tenth with a calculator.], [#math.equation(block: false, alt: "x approximately equals 7.1")[$#h(2.6em) x ≈ 7.1$]],
  [#strong[Step 6. Check] the answer. #linebreak() Check on your own in the Pythagorean Theorem.], [],
  [#strong[Step 7. Answer] the question.], [The pole should be about 7.1 feet tall.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The sun casts a shadow from a flag pole. The height of the flag pole is three times the length of its shadow. The distance between the end of the shadow and the top of the flag pole is 20 feet. Find the length of the shadow and the length of the flag pole. Round to the nearest tenth of a foot.

#solutionbox[
The length of the shadow is 6.3 feet and the length of the flag pole is 18.9 ft.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The distance between opposite corners of a rectangular field is four more than the width of the field. The length of the field is twice its width. Find the distance between the opposite corners. Round to the nearest tenth.

#solutionbox[
The distance to the opposite corner is 3.2.
]
]

#examplebox("Example 4")[][
Mike wants to put 150 square feet of artificial turf in his front yard. This is the maximum area of artificial turf allowed by his homeowners association. He wants to have a rectangular area of turf with length one foot less than three times the width. Find the length and width. Round to the nearest tenth of a foot.

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[Step 1. Read] the problem. Draw a picture.], [#figure(figph[A rectangle is shown with its dimensions labeled algebraically: the width is 'w' and the length is '3w-1'.], alt: "A rectangle is shown with its dimensions labeled algebraically: the width is 'w' and the length is '3w-1'.", caption: none)]),
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for the length and width.],
  [#strong[Step 3. Name] what we are looking for.], [Let #math.equation(block: false, alt: "w equals")[$w =$] the width of the rectangle. #linebreak() #math.equation(block: false, alt: "3 w minus 1 equals")[$3 w − 1 =$] the length of the rectangle],
  [#strong[Step 4. Translate] into an equation.  #linebreak() We know the area. Write the formula for the area of a rectangle.], [#figure(figph[The mathematical formula for calculating area: A = L \* W.], alt: "The mathematical formula for calculating area: A = L * W.", caption: none)],
  [#strong[Step 5. Solve] the equation. Substitute in the values.], [#figure(figph[A mathematical equation is displayed on a white background, which reads 150 = (3w - 1)w. The numbers and variables are in a dark gray font.], alt: "A mathematical equation is displayed on a white background, which reads 150 = (3w - 1)w. The numbers and variables are in a dark gray font.", caption: none)],
  [Distribute.], [#figure(figph[A mathematical equation is displayed on a white background: 150 = 3w^2 - W. The numbers and variables are in a dark gray font.], alt: "A mathematical equation is displayed on a white background: 150 = 3w^2 - W. The numbers and variables are in a dark gray font.", caption: none)],
  [This is a quadratic equation, rewrite it in standard form.], [#figure(figph[Two quadratic equations are displayed: the general form ax^2 + bx + c = 0, and a specific example 3w^2 - w - 150 = 0.], alt: "Two quadratic equations are displayed: the general form ax^2 + bx + c = 0, and a specific example 3w^2 - w - 150 = 0.", caption: none)],
  [Solve the equation using the Quadratic Formula.], [],
  [Identify the #emph[a, b, c] values.], [#figure(figph[The image displays the variable assignments a = 3, b = -1, c = -150, written in a horizontal line with different colors for each assignment: 'a' in light blue, 'b' in red, and 'c' in yellow.], alt: "The image displays the variable assignments a = 3, b = -1, c = -150, written in a horizontal line with different colors for each assignment: 'a' in light blue, 'b' in red, and 'c' in yellow.", caption: none)],
  [Write the Quadratic Formula.], [#figure(figph[The quadratic formula with 'w' as the variable, showing w equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.], alt: "The quadratic formula with 'w' as the variable, showing w equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.", caption: none)],
  [Then substitute in the values of #emph[a, b, c].], [#figure(figph[The quadratic formula with substituted values for a (3, blue), b (-1, red), and c (-150, yellow) to solve for W.], alt: "The quadratic formula with substituted values for a (3, blue), b (-1, red), and c (-150, yellow) to solve for W.", caption: none)],
  [Simplify.], [#figure(figph[The image shows a mathematical equation where W is calculated as the fraction of (1 plus or minus the square root of (1 + 1800)) all divided by 6.], alt: "The image shows a mathematical equation where W is calculated as the fraction of (1 plus or minus the square root of (1 + 1800)) all divided by 6.", caption: none)

 #linebreak() #figure(figph[A mathematical equation shows W equals the fraction of one plus or minus the square root of 1801, all divided by six.], alt: "A mathematical equation shows W equals the fraction of one plus or minus the square root of 1801, all divided by six.", caption: none)],
  [Rewrite to show two solutions.], [#figure(figph[Two mathematical expressions for 'w' are displayed, showing the solutions to a quadratic equation involving a square root and division by six: w = (1 + √1801)/6 and w = (1 - √1801)/6.], alt: "Two mathematical expressions for 'w' are displayed, showing the solutions to a quadratic equation involving a square root and division by six: w = (1 + √1801)/6 and w = (1 - √1801)/6.", caption: none)],
  [Approximate the answers using a calculator. #linebreak() We eliminate the negative solution for the width.], [#figure(figph[Mathematical steps showing the calculation of width (w ≈ 7.2) and length (L ≈ 3w - 1, resulting in L ≈ 20.6), with a negative 'w' value crossed out as invalid.], alt: "Mathematical steps showing the calculation of width (w ≈ 7.2) and length (L ≈ 3w - 1, resulting in L ≈ 20.6), with a negative 'w' value crossed out as invalid.", caption: none)],
  [#strong[Step 6. Check] the answer. #linebreak() Make sure that the answers make sense.], [],
  [#strong[Step 7. Answer] the question.], [The width of the rectangle is approximately 7.2 feet and the length 20.6 feet.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
The length of a 200 square foot rectangular vegetable garden is four feet less than twice the width. Find the length and width of the garden. Round to the nearest tenth of a foot.

#solutionbox[
The width of the garden is 11 feet and the length is 18 feet.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A rectangular tablecloth has an area of 80 square feet. The width is 5 feet shorter than the length. What are the length and width of the tablecloth? Round to the nearest tenth of a foot.

#solutionbox[
The width of the tablecloth is 6.8 feet and the length is 11.8 feet.
]
]

The height of a projectile shot upwards is modeled by a quadratic equation. The initial velocity, #math.equation(block: false, alt: "v sub 0")[$v_(0)$], propels the object up until gravity causes the object to fall back down.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Projectile Motion]

The height in feet, #math.equation(block: false, alt: "h")[$h$], of an object shot upwards into the air with initial velocity, #math.equation(block: false, alt: "v sub 0")[$v_(0)$], after #math.equation(block: false, alt: "t")[$t$] seconds is given by the formula:

#math.equation(block: true, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$]
]

We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.

#examplebox("Example 5")[][
A firework is shot upwards with initial velocity 130 feet per second. How many seconds will it take to reach a height of 260 feet? Round to the nearest tenth of a second.

#solutionbox[
#figure(table(
  columns: 3, align: left, inset: 6pt,
  [#strong[Step 1. Read] the problem.], [], [],
  [#strong[Step 2. Identify] what we are looking for.], [We are looking for the number of seconds, which is time.], [],
  [#strong[Step 3. Name] what we are looking for.], [Let #math.equation(block: false, alt: "t equals")[$t =$] the number of seconds.], [],
  [#strong[Step 4. Translate] into an equation.], [Use the formula.], [],
  [], [], [#math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t #h(1.2em)$]],
  [#strong[Step 5. Solve] the equation.  #linebreak() We know the velocity #math.equation(block: false, alt: "v sub 0")[$v_(0)$] is 130 feet per second.], [], [],
  [The height is 260 feet. Substitute the values.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '260 = -16t^2 + 130t'.], alt: "A mathematical equation is displayed on a white background, which reads '260 = -16t^2 + 130t'.", caption: none)], [],
  [This is a quadratic equation, rewrite it in standard form.], [#figure(figph[Two quadratic equations are shown, with the general form ax^2 + bx + c = 0 in red text, followed by a specific example, 16t^2 - 130t + 260 = 0, in black text.], alt: "Two quadratic equations are shown, with the general form ax^2 + bx + c = 0 in red text, followed by a specific example, 16t^2 - 130t + 260 = 0, in black text.", caption: none)], [],
  [Solve the equation using the Quadratic Formula.], [], [],
  [Identify the #emph[a, b, c] values.], [#figure(figph[The image displays the variables and their assigned numerical values: a = 16, b = -130, and c = 260, presented in different colors on a white background.], alt: "The image displays the variables and their assigned numerical values: a = 16, b = -130, and c = 260, presented in different colors on a white background.", caption: none)], [],
  [Write the Quadratic Formula.], [#figure(figph[The quadratic formula, t equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a, is displayed on a white background.], alt: "The quadratic formula, t equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a, is displayed on a white background.", caption: none)], [],
  [Then substitute in the values of #emph[a, b, c].], [#figure(figph[A mathematical equation for 't' using the quadratic formula is displayed, with 'a' as 16 (light blue), 'b' as -130 (red), and 'c' as 260 (yellow), to find the roots of a quadratic equation.], alt: "A mathematical equation for 't' using the quadratic formula is displayed, with 'a' as 16 (light blue), 'b' as -130 (red), and 'c' as 260 (yellow), to find the roots of a quadratic equation.", caption: none)], [],
  [Simplify.], [#figure(figph[A mathematical equation for 't' involves 130 plus or minus the square root of the difference between 16,900 and 16,640, with the entire expression divided by 32. This formula is commonly used in solving quadratic equations or statistical problems.], alt: "A mathematical equation for 't' involves 130 plus or minus the square root of the difference between 16,900 and 16,640, with the entire expression divided by 32. This formula is commonly used in solving quadratic equations or statistical problems.", caption: none)

 #linebreak() #figure(figph[A mathematical equation is displayed on a white background: t = (130 ± √260) / 32.], alt: "A mathematical equation is displayed on a white background: t = (130 ± √260) / 32.", caption: none)], [],
  [Rewrite to show two solutions.], [#figure(figph[Two mathematical solutions for 't' are presented, each as a fraction: (130 + sqrt(260))/32 and (130 - sqrt(260))/32.], alt: "Two mathematical solutions for 't' are presented, each as a fraction: (130 + sqrt(260))/32 and (130 - sqrt(260))/32.", caption: none)], [],
  [Approximate the answers with a calculator.], [#math.equation(block: false, alt: "t approximately equals 4.6")[$#h(3em) t ≈ 4.6$] seconds, #math.equation(block: false, alt: "t approximately equals 3.6")[$t ≈ 3.6$]], [],
  [#strong[Step 6. Check] the answer. #linebreak() The check is left to you.], [], [],
  [#strong[Step 7. Answer] the question.], [The firework will go up and then fall back down. #linebreak() As the firework goes up, it will reach 260 feet after #linebreak() approximately 3.6 seconds. It will also pass that #linebreak() height on the way down at 4.6 seconds. #linebreak() #figure(figph[A vivid firework traces an upward arc, exploding into a bright, starburst-like display of orange and yellow sparks against a clean white background.], alt: "A vivid firework traces an upward arc, exploding into a bright, starburst-like display of orange and yellow sparks against a clean white background.", caption: none)], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
An arrow is shot from the ground into the air at an initial speed of 108 ft/sec. Use the formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when the arrow will be 180 feet from the ground. Round the nearest tenth of a second.

#solutionbox[
The arrow will reach 180 on its way up in 3 seconds, and on the way down in 3.8 seconds.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
A man throws a ball into the air with a velocity of 96 ft/sec. Use the formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when the height of the ball will be 48 feet. Round to the nearest tenth of a second.

#solutionbox[
The ball will reach 48 feet on its way up in .6 seconds and on the way down in 5.5 seconds.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with solving word problems using the quadratic equation:

- #link("https://openstax.org/l/25Quadproblem")[General Quadratic Word Problems]
- #link("https://openstax.org/l/25Projectile")[Word problem: Solve a projectile problem using a quadratic equation]
]


=== Key Concepts

- #strong[Area of a Triangle] For a triangle with base, #math.equation(block: false, alt: "b")[$b$], and height, #math.equation(block: false, alt: "h")[$h$], the area, #math.equation(block: false, alt: "A")[$A$], is given by the formula: #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] #linebreak() 
#figure(figph[A diagram illustrating a triangle with its base labeled 'b' and its height labeled 'h'. The height 'h' is drawn from the apex perpendicular to the base 'b', dividing the original triangle into two smaller right-angled triangles.], alt: "A diagram illustrating a triangle with its base labeled 'b' and its height labeled 'h'. The height 'h' is drawn from the apex perpendicular to the base 'b', dividing the original triangle into two smaller right-angled triangles.", caption: none)
- #strong[Pythagorean Theorem] In any right triangle, where #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are the lengths of the legs, and #math.equation(block: false, alt: "c")[$c$] is the length of the hypothenuse, #math.equation(block: false, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$] #linebreak() 
#figure(figph[A right-angled triangle with sides labeled a, b, and hypotenuse c, commonly used to demonstrate the Pythagorean theorem.], alt: "A right-angled triangle with sides labeled a, b, and hypotenuse c, commonly used to demonstrate the Pythagorean theorem.", caption: none)
- #strong[Projectile motion] The height in feet, #math.equation(block: false, alt: "h")[$h$], of an object shot upwards into the air with initial velocity, #math.equation(block: false, alt: "v sub 0")[$v_(0)$], after #math.equation(block: false, alt: "t")[$t$] seconds can be modeled by the formula: #linebreak() 
#math.equation(block: true, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$]


==== Practice Makes Perfect

#strong[Solve Applications of the Quadratic Formula]

In the following exercises, solve by using methods of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth.

The product of two consecutive odd numbers is 255. Find the numbers.

#solutionbox[
Two consecutive odd numbers whose product is 255 are 15 and 17, and #math.equation(block: false, alt: "−15")[$−15$] and #math.equation(block: false, alt: "−17")[$−17$].
]

The product of two consecutive even numbers is 360. Find the numbers.

The product of two consecutive even numbers is 624. Find the numbers.

#solutionbox[
Two consecutive even numbers whose product is 624 are 24 and 26, and #math.equation(block: false, alt: "−26")[$−26$] and #math.equation(block: false, alt: "−24")[$−24$].
]

The product of two consecutive odd numbers is 1023. Find the numbers.

The product of two consecutive odd numbers is 483. Find the numbers.

#solutionbox[
Two consecutive odd numbers whose product is 483 are 21 and 23, and #math.equation(block: false, alt: "−21")[$−21$] and #math.equation(block: false, alt: "−23")[$−23$].
]

The product of two consecutive even numbers is 528. Find the numbers.

A triangle with area 45 square inches has a height that is two less than four times the width. Find the height and width of the triangle.

#solutionbox[
The width of the triangle is 5 inches and the height is 18 inches.
]

The width of a triangle is six more than twice the height. The area of the triangle is 88 square yards. Find the height and width of the triangle.

The hypotenuse of a right triangle is twice the length of one of its legs. The length of the other leg is three feet. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

#solutionbox[
The leg of the right triangle is 1.7 feet and the hypotenuse is 3.4 feet.
]

The hypotenuse of a right triangle is 10 cm long. One of the triangle’s legs is three times the length of the other leg. Find the lengths of the three sides of the triangle. Round to the nearest tenth.

A farmer plans to fence off sections of a rectangular corral. The diagonal distance from one corner of the corral to the opposite corner is five yards longer than the width of the corral. The length of the corral is three times the width. Find the length of the diagonal of the corral. Round to the nearest tenth.

#figure(figph[The image shows rectangle with the long sides horizontal. A diagonal line runs from the top left corner of the rectangle to the bottom right corner.], alt: "The image shows rectangle with the long sides horizontal. A diagonal line runs from the top left corner of the rectangle to the bottom right corner.", caption: none)

#solutionbox[
The length of the diagonal of the fence is 7.3 yards.
]

Nautical flags are used to represent letters of the alphabet. The flag for the letter O consists of a yellow right triangle and a red right triangle which are sewn together along their hypotenuse to form a square. The adjoining side of the two triangles is three inches longer than a side of the flag. Find the length of the side of the flag.

#figure(figph[The image shows a square with a diagonal line running from the top left corner to the bottom right corner. The diagonal splits the square into two right triangles. The lower triangle is red and the upper triangle is yellow.], alt: "The image shows a square with a diagonal line running from the top left corner to the bottom right corner. The diagonal splits the square into two right triangles. The lower triangle is red and the upper triangle is yellow.", caption: none)

The length of a rectangular driveway is five feet more than three times the width. The area is 350 square feet. Find the length and width of the driveway.

#solutionbox[
The width of the driveway is 10 feet and its length is 35 feet.
]

A rectangular lawn has area 140 square yards. Its width that is six less than twice the length. What are the length and width of the lawn?

A firework rocket is shot upward at a rate of 640 ft/sec. Use the projectile formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when the height of the firework rocket will be 1200 feet.

#solutionbox[
The rocket will reach 1,200 feet on its way up in 2 seconds and on the way down in 38 seconds.
]

An arrow is shot vertically upward at a rate of 220 feet per second. Use the projectile formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when height of the arrow will be 400 feet.


==== Everyday Math

A bullet is fired straight up from a BB gun with initial velocity 1120 feet per second at an initial height of 8 feet. Use the formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t plus 8")[$h = −16 t^(2) + v_(0) t + 8$] to determine how many seconds it will take for the bullet to hit the ground. (That is, when will #math.equation(block: false, alt: "h equals 0")[$h = 0$]
?)

#solutionbox[
70 seconds
]

A city planner wants to build a bridge across a lake in a park. To find the length of the bridge, he makes a right triangle with one leg and the hypotenuse on land and the bridge as the other leg. The length of the hypotenuse is 340 feet and the leg is 160 feet. Find the length of the bridge.

#figure(figph[The image shows a right triangle with a horizontal side stretching across a lake, a vertical side on the left labeled a and the hypotenuse connecting the two.], alt: "The image shows a right triangle with a horizontal side stretching across a lake, a vertical side on the left labeled a and the hypotenuse connecting the two.", caption: none)


==== Writing Exercises

Make up a problem involving the product of two consecutive odd integers. Start by choosing two consecutive odd integers. ⓐ What are your integers? ⓑ What is the product of your integers? ⓒ Solve the equation #math.equation(block: false, alt: "n open parenthesis n plus 2 close parenthesis equals p")[$n ( n + 2 ) = p$], where #math.equation(block: false, alt: "p")[$p$] is the product you found in part (b). ⓓ Did you get the numbers you started with?

#solutionbox[
ⓐ answers will vary #linebreak() ⓑ answers will vary ⓒ answers will vary ⓓ answers will vary
]

Make up a problem involving the product of two consecutive even integers. Start by choosing two consecutive even integers. ⓐ What are your integers? ⓑ What is the product of your integers? ⓒ Solve the equation #math.equation(block: false, alt: "n open parenthesis n plus 2 close parenthesis equals p")[$n ( n + 2 ) = p$], where #math.equation(block: false, alt: "p")[$p$] is the product you found in part (b). ⓓ Did you get the numbers you started with?


==== 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 is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “solve applications of the quadratic formula.” The remaining columns are blank.], alt: "This table has two rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the next row reads “solve applications of the quadratic formula.” The remaining columns are blank.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?