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16.E: Review Exercises 2

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Chapter Review Exercises

Solve Quadratic Equations Using the Square Root Property

Exercise 16.E.1 Solve Quadratic Equations of the Form ax2=k Using the Square Root Property

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

  1. y2=144
  2. n280=0
  3. 4a2=100
  4. 2b2=72
  5. r2+32=0
  6. t2+18=0
  7. 23w220=30
  8. 5c2+3=19
Answer

1. y=±12

3. a=±5

5. r=±42i

7. w=±53

Exercise 16.E.2 Solve Quadratic Equations of the Form a(xh)2=k Using the Square Root Property

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

  1. (p5)2+3=19
  2. (u+1)2=45
  3. (x14)2=316
  4. (y23)2=29
  5. (n4)250=150
  6. (4c1)2=18
  7. n2+10n+25=12
  8. 64a2+48a+9=81
Answer

1. p=1,9

3. x=14±34

5. n=4±102

7. n=5±23

Solve Quadratic Equations by Completing the Square

Exercise 16.E.3 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.

  1. x2+22x
  2. m28m
  3. a23a
  4. b2+13b
Answer

1. (x+11)2

3. (a32)2

Exercise 16.E.4 Solve Quadratic Equations Using Completing the Square

In the following exercises, solve by completing the square.

  1. d2+14d=13
  2. y26y=36
  3. m2+6m=109
  4. t212t=40
  5. v214v=31
  6. w220w=100
  7. m2+10m4=13
  8. n26n+11=34
  9. a2=3a+8
  10. b2=11b5
  11. (u+8)(u+4)=14
  12. (z10)(z+2)=28
Answer

1. d=13,1

3. m=3±10i

5. v=7±32

7. m=9,1

9. a=32±412

11. u=6±22

Solve Quadratic Equations of the Form ax2+bx+c=0 by Completing the Square

Exercise 16.E.5 Solve Quadratic Equations of the Form ax2+bx+c=0 by Completing the Square

In the following exercises, solve by completing the square.

  1. 3p218p+15=15
  2. 5q2+70q+20=0
  3. 4y26y=4
  4. 2x2+2x=4
  5. 3c2+2c=9
  6. 4d22d=8
  7. 2x2+6x=5
  8. 2x2+4x=5
Answer

1. p=0,6

3. y=12,2

5. c=13±273

7. x=32±12i

Exercise 16.E.6 Solve Quadratic Equations Using the Quadratic Formula

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

  1. 4x25x+1=0
  2. 7y2+4y3=0
  3. r2r42=0
  4. t2+13t+22=0
  5. 4v2+v5=0
  6. 2w2+9w+2=0
  7. 3m2+8m+2=0
  8. 5n2+2n1=0
  9. 6a25a+2=0
  10. 4b2b+8=0
  11. u(u10)+3=0
  12. 5z(z2)=3
  13. 18p215p=120
  14. 25q2+310q=110
  15. 4c2+4c+1=0
  16. 9d212d=4
Answer

1. x=14,1

3. r=6,7

5. v=1±218

7. m=4±103

9. a=512±2312i

11. u=5±21

13. p=4±55

15. c=12

Exercise 16.E.7 Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

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

    1. 9x26x+1=0
    2. 3y28y+1=0
    3. 7m2+12m+4=0
    4. 5n2n+1=0
    1. 5x27x8=0
    2. 7x210x+5=0
    3. 25x290x+81=0
    4. 15x28x+4=0
Answer

1.

  1. 1
  2. 2
  3. 2
  4. 2
Exercise 16.E.8 Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

    1. 16r28r+1=0
    2. 5t28t+3=9
    3. 3(c+2)2=15
    1. 4d2+10d5=21
    2. 25x260x+36=0
    3. 6(5v7)2=150
Answer

1.

  1. Factor
  2. Quadratic Formula
  3. Square Root

Solve Equations in Quadratic Form

Exercise 16.E.9 Solve Equations in Quadratic Form

In the following exercises, solve.

  1. x414x2+24=0
  2. x4+4x232=0
  3. 4x45x2+1=0
  4. (2y+3)2+3(2y+3)28=0
  5. x+3x28=0
  6. 6x+5x6=0
  7. x2310x13+24=0
  8. x+7x12+6=0
  9. 8x22x13=0
Answer

1. x=±2,x=±23

3. x=±1,x=±12

5. x=16

7. x=64,x=216

9. x=2,x=43

Solve Applications of Quadratic Equations

Exercise 16.E.10 Solve Applications Modeled by Quadratic Equations

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

  1. Find two consecutive odd numbers whose product is 323.
  2. Find two consecutive even numbers whose product is 624.
  3. 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.
  4. 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.
  5. 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.

A rectangle is shown is a right triangle in the corner. The hypotenuse of the triangle is 5 feet, the longer leg is 2 times s and the shorter leg is s.
Figure 9.E.1

6. 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.

7. 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.

8. For Sophia’s graduation party, several tables of the same width will be arranged end to end to give 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.

9. A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula h=16t2+v0t to determine when the ball will be 384 feet from the ground. Round to the nearest tenth.

10. The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?

11. Ezra kayaked up the river and then back in a total time of 6 hours. The trip was 4 miles each way and the current was difficult. If Roy kayaked at a speed of 5 mph, what was the speed of the current?

12. Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

Answer

2. Two consecutive even numbers whose product is 624 are 24 and 26, and 24 and 26.

4. The height is 14 inches and the width is 10 inches.

6. The length of the diagonal is 3.6 feet.

8. The width of the serving table is 4.7 feet and the length is 16.1 feet.

Four tables arranged end-to-end are shown. Together, they have an area of 75 feet. The short side measures w and the long side measures 3 times w plus 2.
Figure 9.E.2

10. The speed of the wind was 30 mph.

12. One man takes 3 hours and the other man 6 hours to finish the repair alone.

Graph Quadratic Functions Using Properties

Exercise 16.E.11 Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

  1. Graph y=x22
  2. Graph y=x2+3
Answer

2.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (negative 2, negative 1) and (2, negative 1).
Figure 9.E.3
Exercise 16.E.12 Recognize the Graph of a Quadratic Function

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

    1. y=3x2+3x1
    2. y=5x2+6x+3
    1. y=x2+8x1
    2. y=4x27x+1
Answer

2.

  1. Up
  2. Down
Exercise 16.E.13 Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find

  1. The equation of the axis of symmetry
  2. The vertex
    1. y=x2+6x+8
    2. y=2x28x+1
Answer

2. x=2 ; (2,7)

Exercise 16.E.14 Find the Intercepts of a Parabola

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

  1. y=x24x+5
  2. y=x28x+15
  3. y=x24x+10
  4. y=5x230x46
  5. y=16x28x+1
  6. y=x2+16x+64
Answer

2. y:(0,15)x:(3,0),(5,0)

4. y:(0,46)x: none 

6. y:(0,64)x:(8,0)

Graph Quadratic Functions Using Properties

Exercise 16.E.15 Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

  1. y=x2+8x+15
  2. y=x22x3
  3. y=x2+8x16
  4. y=4x24x+1
  5. y=x2+6x+13
  6. y=2x28x12
Answer

2.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and a y-intercept of (0, negative 3).
Figure 9.E.4

4.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, 0) and a y-intercept of (0, 1).
Figure 9.E.5

6.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, negative 4) and a y-intercept of (0, negative 12).
Figure 9.E.6
Exercise 16.E.16 Solve Maximum and Minimum Applications

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

  1. y=7x2+14x+6
  2. y=3x2+12x10
Answer

2. The maximum value is 2 when x=2.

Exercise 16.E.17 Solve Maximum and Minimum Applications

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

  1. A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation h=16t2+112t to find how long it will take the ball to reach maximum height, and then find the maximum height.
  2. 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=2x2+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.
An odd-shaped figure is given. 3 sides of a rectangle are attached to the right side of the figure.
Figure 9.E.7
Answer

2. The length adjacent to the building is 90 feet giving a maximum area of 4,050 square feet.

Graph Quadratic Functions Using Transformations

Exercise 16.E.18 Graph Quadratic Functions of the Form f(x)=x2+k

In the following exercises, graph each function using a vertical shift.

  1. g(x)=x2+4
  2. h(x)=x23
Answer

2.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, 0) and other points of (negative 1, negative 2) and (1, negative 2).
Figure 9.E.8
Exercise 16.E.19 Graph Quadratic Functions of the Form f(x)=x2+k

In the following exercises, graph each function using a horizontal shift.

  1. f(x)=(x+1)2
  2. g(x)=(x3)2
Answer

2.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (3, 0) and other points of (2, 1) and (4,1).
Figure 9.E.9
Exercise 16.E.20 Graph Quadratic Functions of the Form f(x)=x2+k

In the following exercises, graph each function using transformations.

  1. f(x)=(x+2)2+3
  2. f(x)=(x+3)22
  3. f(x)=(x1)2+4
  4. f(x)=(x4)23
Answer

2.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 3, negative 2) and other points of (negative 5, 2) and (negative 1, 2).
Figure 9.E.10

4.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 3) and other points of (3, negative 2) and (5, negative 2).
Figure 9.E.11
Exercise 16.E.21 Graph Quadratic Functions of the Form f(x)=ax2

In the following exercises, graph each function.

  1. f(x)=2x2
  2. f(x)=x2
  3. f(x)=12x2
Answer

2.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 0) and other points of (negative 1, negative 1) and (1, negative 1).
Figure 9.E.12
Exercise 16.E.22 Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the f(x)=a(xh)2+k form by completing the square.

  1. f(x)=2x24x4
  2. f(x)=3x2+12x+8
Answer

1. f(x)=2(x1)26

Exercise 16.E.23 Graph Quadratic Functions Using Transformations

In the following exercises,

  1. Rewrite each function in f(x)=a(xh)2+k form
  2. Graph it by using transformations
    1. f(x)=3x26x1
    2. f(x)=2x212x5
    3. f(x)=2x2+4x+6
    4. f(x)=3x212x+7
Answer

1.

  1. f(x)=3(x1)24

  2. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 4) and other points of (0, negative 1) and (2, negative 1).
    Figure 9.E.13

3.

  1. f(x)=2(x+1)2+4

  2. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4) and other points of (negative 2, 6) and (0, 6).
    Figure 9.E.14
Exercise 16.E.24 Graph Quadratic Functions Using Transformations

In the following exercises,

  1. Rewrite each function in f(x)=a(xh)2+k form
  2. Graph it using properties
    1. f(x)=3x212x5
    2. f(x)=2x212x+7
Answer

1.

  1. f(x)=3(x+2)2+7

  2. This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 7) and other points of (negative 4, negative 5) and (0, negative 5).
    Figure 9.E.15
Exercise 16.E.25 Find a Quadratic Function From its Graph

In the following exercises, write the quadratic function in f(x)=a(xh)2+k form.


  1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 1) and other points of (negative 2, negative 4) and (0, negative 4).
    Figure 9.E.16

  2. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 4) and other points of (0, 8) and (4, 8).
    Figure 9.E.17
Answer

1. f(x)=(x+1)25

Solve Quadratic Inequalities

Exercise 16.E.26 Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

  1. x2x6>0
  2. x2+4x+30
  3. x2x+20
  4. x2+2x+3<0
Answer

1.


  1. This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (one-half, negative 6 and one-fourth) and other points of (0, negative 6) and (1, negative 6).
    Figure 9.E.18
  2. (,2)(3,)

3.


  1. This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative one-half, 2 and one-fourth) and other points of (negative 2, 0) and (1, 0).
    Figure 9.E.19
  2. [2,1]
Exercise 16.E.27 Solve Quadratic Inequalities Graphically

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

  1. x26x+8<0
  2. x2+x>12
  3. x26x+40
  4. 2x2+7x4>0
  5. x2+x6>0
  6. x22x+40
Answer

1. (2,4)

3. [35,3+5]

5. no solution

Practice Test

Exercise 16.E.28
  1. Use the Square Root Property to solve the quadratic equation 3(w+5)2=27.
  2. Use Completing the Square to solve the quadratic equation a28a+7=23.
  3. Use the Quadratic Formula to solve the quadratic equation 2m25m+3=0.
Answer

1. w=2,w=8

3. m=1,m=32

Exercise 16.E.29

Solve the following quadratic equations. Use any method.

  1. 2x(3x2)1=0
  2. 94y23y+1=0
Answer

2. y=23

Exercise 16.E.30

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

  1. 6p213p+7=0
  2. 3q210q+12=0
Answer

2. 2 complex

Exercise 16.E.31

Solve each equation.

  1. 4x417x2+4=0
  2. y23+2y133=0
Answer

2. y=1,y=27

Exercise 16.E.32

For each parabola, find

  1. Which direction it opens
  2. The equation of the axis of symmetry
  3. The vertex
  4. The x-and y-intercepts
  5. The maximum or minimum value
    1. y=3x2+6x+8
    2. y=x28x+16
Answer

2.

  1. down
  2. x=4
  3. (4,0)
  4. y:(0,16);x:(4,0)
  5. minimum value of 4 when x=0
Exercise 16.E.33

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

  1. f(x)=x2+6x+9
  2. f(x)=2x2+8x+4
Answer

2.

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, 12) and other points of (0, 4) and (4, 4).
Figure 9.E.20
Exercise 16.E.34

In the following exercises, graph each function using transformations.

  1. f(x)=(x+3)2+2
  2. f(x)=x24x1
Answer

2.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 5) and other points of (0, negative 1) and (4, negative 1).
Figure 9.E.21

Exercise 16.E.35

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

  1. x26x80
  2. 2x2+x10>0
Answer

2. (,52)(2,)

Exercise 16.E.36

Model the situation with a quadratic equation and solve by any method.

  1. Find two consecutive even numbers whose product is 360.
  2. 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.)
Answer

2. A water balloon is launched upward at the rate of 86 ft/sec. Using the formula h=16t2+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.


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