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Chapter 10 Review Exercises

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Chapter 10 Review Exercises

10.1 Solve Quadratic Equations Using the Square Root Property

In the following exercises, solve using the Square Root Property.

Exercise 1

x2=100

Answer

x=±10

Exercise 2

y2=144

Exercise 3

m240=0

Answer

m=±210

Exercise 4

n280=0

Exercise 5

4a2=100

Answer

a=±5

Example 6

2b2=72

Exercise 7

r2+32=0

Answer

no solution

Exercise 8

t2+18=0

Exercise 9

43v2+4=28

Answer

v=±32

Exercise 10

23w220=30

Exercise 11

5c2+3=19

Answer

c=±455

Exercise 12

3d26=43

In the following exercises, solve using the Square Root Property.

Exercise 13

(p5)2+3=19

Answer

p=1, 9

Exercise 14

(q+4)2=9

Exercise 15

(u+1)2=45

Answer

u=1±35

Exercise 16

(z5)2=50

Exercise 17

(x14)2=316

Answer

x=14±34

Exercise 18

(y23)2=29

Exercise 19

(m7)2+6=30

Answer

m=7±26

Exercise 20

(n4)250=150

Exercise 21

(5c+3)2=20

Answer

no solution

Exercise 22

(4c1)2=18

Exercise 23

m26m+9=48

Answer

m=3±43

Exercise 24

n2+10n+25=12

Exercise 25

64a2+48a+9=81

Answer

a=−32, 34

Exercise 26

4b228b+49=25

10.2 Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

Exercise 26

x2+22x

Answer

(x+11)2

Exercise 27

y2+6y

Exercise 28

m28m

Answer

(m4)2

Exercise 29

n210n

Exercise 30

a23a

Answer

(a32)2

Exercise 31

b2+13b​​​​

Exercise 32

p2+45p

Answer

(p+25)2​​​​​

Exercise 33

q213q

In the following exercises, solve by completing the square.

Exercise 34

c2+20c=21

Answer

c=1, −21

Exercise 35

d2+14d=13​​​​​

Exercise 36

x24x=32

Answer

x=−4, 8

Exercise 37

y216y=36​​​​

Exercise 38

r2+6r=100

Answer

no solution​​​​​

Exercise 39

t212t=40​​​​​

Exercise 40

v214v=31

Answer

v=7±32​​​​​

Exercise 41

w220w=100​​​​​

Exercise 42

m2+10m4=13

Answer

m=9,1​​​​​

Exercise 43

n26n+11=34​​​​​

Exercise 44

a2=3a+8

Answer

a=32±412​​​

Exercise 45

b2=11b5

Exercise 46

(u+8)(u+4)=14

Answer

u=6±22​​​​

Exercise 47

(z10)(z+2)=28​​​​​

Exercise 48

3p218p+15=15

Answer

p=0, 6

Exercise 49

5q2+70q+20=0​​​​​

Exercise 50

4y26y=4

Answer

y=12,2​​​​​

Exercise 51

2x2+2x=4​​​​​

Exercise 52

3c2+2c=9

Answer

c=13±273​​​​​​

Exercise 53

4d22d=8

10.3 Solve Quadratic Equations Using the Quadratic Formula

In the following exercises, solve by using the Quadratic Formula.

Exercise 54

4x25x+1=0

Answer

x=14,1

Exercise 55

7y2+4y3=0

Exercise 56

r2r42=0

Answer

r=6,7

Exercise 57

t2+13t+22=0

Exercise 58

4v2+v5=0

Answer

v=54,1

Exercise 59

2w2+9w+2=0​​​​​

Exercise 60

3m2+8m+2=0

Answer

m=4±103​​​​​​

Exercise 61

5n2+2n1=0

Exercise 62

6a25a+2=0

Answer

no real solution​​​

Exercise 63

4b2b+8=0

Exercise 64

u(u10)+3=0

Answer

u=5±22

Exercise 65

5z(z2)=3​​​​​​​​​​​​​

Exercise 66

18p215p=120

Answer

p=4±65​​​​​​​

Exercise 67

25q2+310q=110​​​​​​​

Exercise 68

4c2+4c+1=0

Answer

c=−\frac{1}{2}

Exercise \PageIndex{69}

9d^2−12d=−4

In the following exercises, determine the number of solutions to each quadratic equation.

Exercise \PageIndex{70}
  1. 9x^2−6x+1=0
  2. 3y^2−8y+1=0
  3. 7m^2+12m+4=0
  4. 5n^2−n+1=0
Answer
  1. 1
  2. 2
  3. 2
  4. none
Exercise \PageIndex{71}
  1. 5x^2−7x−8=0
  2. 7x^2−10x+5=0
  3. 25x^2−90x+81=0
  4. 15x^2−8x+4=0

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.

Exercise \PageIndex{72}
  1. 16r^2−8r+1=0
  2. 5t^2−8t+3=93(c+2)^2=15
Answer
  1. factor
  2. Quadratic Formula
  3. square root
Exercise \PageIndex{73}
  1. 4d^2+10d−5=21
  2. 25x^2−60x+36=0
  3. 6(5v−7)^2=150

10.4 Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.

Exercise \PageIndex{74}

Find two consecutive odd numbers whose product is 323.

Answer

Two consecutive odd numbers whose product is 323 are 17 and 19, and −17 and −19.

Exercise \PageIndex{75}

Find two consecutive even numbers whose product is 624.

Exercise \PageIndex{76}

A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base.

Answer

The height of the banner is 13 cm and the length of the side is 54 cm.​​​​​​

Exercise \PageIndex{77}

Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.​​​​​​​

Exercise \PageIndex{78}

A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth.

The image shows a rectangular pathway with a right inlaid in the lower left corner. The right angle of the triangle overlays the lower left corner of the rectangle. The left leg of the right triangle overlays the left side of the rectangle and the hypotenuse of the right triangle runs from the upper left corner of the rectangle to a point on the bottom of the rectangle.

Answer

The lengths of the sides of the mosaic are 2.2 and 4.4 feet.​​​​​

Exercise \PageIndex{79}

A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.​​​​​​​

Exercise \PageIndex{80}

The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth.

Answer

The width of the front walk is 8.1 feet and its length is 30.8 feet.​​​​

Exercise \PageIndex{81}

For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth.

The image shows four rectangular tables placed side by side to create one large table.​​​​

Exercise \PageIndex{82}

A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula h=−16t^2+v_{0}t to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

Answer

The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds.​​​​

Exercise \PageIndex{83}

A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula h=−16t^2+v_{0}t to determine when the bullet will reach 800 feet. Round to the nearest tenth.​​​​​

10.5 Graphing Quadratic Equations in Two Variables

In the following exercises, graph by plotting point.

Exercise \PageIndex{84}

Graph y=x^2−2

Answer

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, -2) and goes through the point (1, -1).

Exercise \PageIndex{85}

Graph y=−x^2+3

​​​​​​​In the following exercises, determine if the following parabolas open up or down.

Exercise \PageIndex{86}

y=−3x^2+3x−1

Answer

down​​​​​​

Exercise \PageIndex{87}

y=5x^2+6x+3​​​

Exercise \PageIndex{88}

y=x^2+8x−1

Answer

up​​​​

Exercise \PageIndex{89}

Ay=−4x^2−7x+1

​​​​​​In the following exercises, find

  1. the axis of symmetry and,
  2. the vertex.
Exercise \PageIndex{90}

y=−x^2+6x+8

Answer
  1. x=3
  2. (3,17)​​​​
Exercise \PageIndex{91}

y=2x^2−8x+1

​​​​​​​In the following exercises, find the x- and y-intercepts.

Exercise \PageIndex{92}

y=x^2−4x+5

Answer

y:(0,5); x:(5,0), (−1,0)​​​​​​​

Exercise \PageIndex{93}

y=x^2−8x+15​​​​​​

Exercise \PageIndex{94}

y=x^2−4x+10

Answer

y:(0,10); x:none​​​​

Exercise \PageIndex{95}

y=−5x^2−30x−46​​​​

Exercise \PageIndex{96}

y=16x^2−8x+1

Answer

y:(0,1); x:(14,0)​​​​

Exercise \PageIndex{97}

y=x^2+16x+64

​​​​​​​In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.

Exercise \PageIndex{98}

y=x^2+8x+15

Answer

y:(0,15) ;x:(−3,0), (−5,0);
axis: x=−4; vertex:(−4,−1)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 17. The parabola has points plotted at the vertex (-4, -1) and the intercepts (-3, 0), (-5, 0) and (0, 15). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -4.​​​​​

Exercise \PageIndex{99}

y=x^2−2x−3​​​​​

Exercise \PageIndex{100}

y=−x^2+8x−16

Answer

y:(0,−16); x:(4,0);
axis: x=4; vertex:(4,0)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -15 to 12. The y-axis of the plane runs from -20 to 2. The parabola has points plotted at the vertex (4, 0) and the intercept (0, -16). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.​​​​

Exercise \PageIndex{101}

y=4x^2−4x+1​​​​​

Exercise \PageIndex{102}

y=x^2+6x+13

Answer

y:(0,13); x:none;
axis: x=−3; vertex:(−3,4)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 18. The parabola has points plotted at the vertex (-3, 4) and the intercept (0, 13). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.​​​​​​

Exercise \PageIndex{103}

y=−2x^2−8x−12​​​​​

Exercise \PageIndex{104}

y=−4x^2+16x−11

Answer

y:(0,−11); x:(3.1,0),(0.9,0);
axis: x=2; vertex:(2,5)

This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (2, 5) and the intercepts (3.1, 0) and (0.9, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.​​​​​​

Exercise \PageIndex{105}

y=x^2+8x+10

​​​​​​​In the following exercises, find the minimum or maximum value.

Exercise \PageIndex{106}

y=7x^2+14x+6

Answer

The minimum value is −1 when x=−1.

Exercise \PageIndex{107}

y=−3x^2+12x−10

In the following exercises, solve. Rounding answers to the nearest tenth.

Exercise \PageIndex{108}

A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation h=−16t^2+112t to find how long it will take the ball to reach maximum height, and then find the maximum height.

Answer

In 3.5 seconds the ball is at its maximum height of 196 feet.​​​​​​​​​​​​​​

Exercise \PageIndex{109}

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation A=−2x^2+180x gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.​​​​​

Practice Test

Exercise \PageIndex{110}

Use the Square Root Property to solve the quadratic equation: 3(w+5)^2=27.

Answer

w=−2, −8

Exercise \PageIndex{111}

Use Completing the Square to solve the quadratic equation: a^2−8a+7=23​​​​​​​

Exercise \PageIndex{112}

Use the Quadratic Formula to solve the quadratic equation: 2m^2−5m+3=0.

Answer

m=1, 32

​​​​​​​Solve the following quadratic equations. Use any method.​​​​​​​

Exercise \PageIndex{113}

8v^2+3=35​​​​​​

Exercise \PageIndex{114}

3n^2+8n+3=0

Answer

n=\frac{−4\pm\sqrt{7}}{3}

Exercise \PageIndex{115}

2b^2+6b−8=0​​​​​

Exercise \PageIndex{116}

x(x+3)+12=0

Answer

no real solution

Exercise \PageIndex{117}

\frac{4}{3}y^2−4y+3=0

​​​​​​​Use the discriminant to determine the number of solutions of each quadratic equation.

Exercise \PageIndex{118}

6p^2−13p+7=0

Answer

2

Exercise \PageIndex{119}

3q^2−10q+12=0

Solve by factoring, the Square Root Property, or the Quadratic Formula.

Exercise \PageIndex{120}

Find two consecutive even numbers whose product is 360.

Answer

Two consecutive even number are −20 and −18 and 18 and 20.

Exercise \PageIndex{121}

The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)

For each parabola, find

  1. which ways it opens,
  2. the axis of symmetry,
  3. the vertex,
  4. the x- and y-intercepts, and
  5. the maximum or minimum value.
Exercise \PageIndex{122}

y=3x^2+6x+8

Answer
  1. up
  2. x=−1
  3. (−1,5)
  4. y:(0,8);x:none; y:(0,8)
  5. minimum value of 5 when x=−1.
Exercise \PageIndex{123}

y=x^2−4

Exercise \PageIndex{124}

y=x^2+10x+24

Answer
  1. up
  2. x=−5
  3. (−5,−1)
  4. y:(0,24); x:(−6,0), (−4,0)
  5. minimum value of −5 when x=−1​​​​​​
Exercise \PageIndex{125}

y=−3x^2+12x−8

Exercise \PageIndex{126}

y=−x^2−8x+16

Answer
  1. down
  2. x=−4
  3. (−4,32)
  4. y;(0,16); x:(−9.7,0),(1.7,0)
  5. maximum value of 32 when x=−4

​​​​​​​Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.​​​​​​​

Exercise \PageIndex{127}

y=2x^2+6x+2​​​​​​​

Exercise \PageIndex{128}

y=16x^2+24x+9

Answer

y:(0,9); x:(−34,0)
axis: x=−\frac{3}{4}; vertex:(−\frac{3}{4},0)

This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (3 fourths, 0) and the intercept (0, 9). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 fourths.

​​​​​​​Solve.

Exercise \PageIndex{129}

A water balloon is launched upward at the rate of 86 ft/sec. Using the formula h=−16t^2+86t, find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.


This page titled Chapter 10 Review Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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