Algebra 2

Problem 1 Solving a Linear-Quadratic System by Graphing

Multiple Choice  W​ hich numbers are y-values of the solutions y = − x 2 + 5x + 6
of the system of equations? ey = x +6

How can you graph 4 only 6 only 4 and 6 6 and 10
these two equations?
Use slope-intercept Graph the equations. Find their intersections.
form to graph the linear
equation. Make a table The solutions appear to be (0, 6) and (4, 10). 12 y
of values to graph the 10
quadratic equation. Check y = - x 2 + 5x + 6 y=x+6 8 (4, 10)
6 ≟ - (0)2 + 5(0) + 6 6≟0 + 6 6 (0, 6)

6=6 ✔ 6=6 ✔

y = - x 2 + 5x + 6 y=x+6

10 ≟ -(4)2 + 5(4) + 6 10 ≟ 4 + 6

10 = 10 ✔ 10 = 10 ✔

The y-values of the solutions are 6 and 10, choice D. x
8
Ϫ2 O 2 4

Got It? 1. What is the solution of the system? y = x2 + 6x + 9
ey = x+3

Problem 2 Solving a Linear-Quadratic System Using Substitution

What is the solution of the system of equations? y = −x2 − x + 6
ey = x+3

Substitute x + 3 for y in   x + 3 = −x2 − x + 6
the quadratic equation.
Write in standard form. x2 + 2x − 3 = 0

Factor. Solve for x.   (x − 1)(x + 3) = 0
  x = 1 or x = −3
Substitute for x in  x=1 S y=1+3=4
y = x + 3.   x = −3 S y = −3 + 3 = 0
  The solutions are (1, 4) and (−3, 0).

Got It? 2. What is the solution of the system? e y = -x2 - 3x + 10
y = x+5

Lesson 4-9  Quadratic Systems 259

You can solve quadratic–quadratic systems using the same methods you used for
linear–quadratic systems.

Problem 3 Solving a Quadratic System of Equations

What is the solution of the system? y = − x2 − x + 12
ey = x2 + 7x + 12
Which variable
should you Method 1 Use substitution.
substitute for?
You can substitute for Substitute y = -x 2 - x + 12 for y in the second equation. Solve for x.
either variable, but
substituting for y results   - x 2 - x + 12 = x 2 + 7x + 12 Substitute for y.
in a simple equation.
  - 2x 2 - 8x = 0 Write in standard form.

  - 2x(x + 4) = 0 Factor.

  x = 0 or x = - 4 Solve for x.

Substitute each value of x into either equation. Solve for y.

  y = x 2 + 7x + 12 y = x 2 + 7x + 12

y = (0)2 + 7(0) + 12 y = ( -4)2 + 7( -4) + 12

y = 0 + 0 + 12 = 12 y = 16 - 28 + 12 = 0

The solutions are (0, 12) and ( -4, 0).

Method 2 Graph the equations. Plot1 Plot2 Plot3
\Y1 = –X2–X+12
Use a graphing calculator. \Y2 = X2+7X+12
Define functions Y1 and Y2. \Y3 =
\Y4 =
\Y5 =
\Y6 =
\Y7 =

Use the intersect feature to find the points of intersection.

Intersection Y=0 Intersection Y=12
X=–4 X=0

The solutions are ( -4, 0) and (0, 12).

Got It? 3. What is the solution of each system of equations? x2 -
a. e yy == x-2x-2 +4x5+ 5 b. - x2
e y = 4x + 5
y = -5

260 Chapter 4  Quadratic Functions and Equations

You can use the techniques for solving a linear system of inequalities to solve a
quadratic system of inequalities.

Problem 4 Solving a Quadratic System of Inequalities

What is the solution of the system of inequalities? y * − x2 − 9x − 2
ey + x2 − 2
How can you find the
solution? 20 y Graph the first 20 y Graph the second
Graph each inequality 16 inequality. 16 inequality.
and find the region
where the graphs
overlap.

Ϫ14 Ϫ10 Ϫ6 Ϫ2 Ox Ϫ14 Ϫ10 Ϫ6 Ϫ2 O Identify where the
48 4 graphs overlap.

x
8

The solution of this system is the region where the graphs overlap. The region contains
no boundary points.

Got It? 4. a. What is the solution of this system of inequalities? e y … - x2 - 4x + 3
y 7 x2 + 3

b. Reasoning  How many solutions can a system of inequalities have?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Solve each system by substitution.
PRACTICES

6. Compare and Contrast  ​How are solving systems

y = x2 - 2x + 3 of two linear equations or inequalities and solving
y = x+1
1. e systems of two quadratic equations or inequalities

alike? How are they different?

2. e y = 2x2 - 5x + 2 7. Reasoning  ​How many points of intersection can
y = x- 2
graphs of the following types of functions have? Draw
3. e y = x2 - 3x - 3 5
y = - 2x2 - x + graphs to justify your answers.
a. a linear function and a quadratic function
Solve each system by graphing. b. two quadratic functions
c. a quadratic function and an absolute value
4. e y 7 2x2 + x + 3 1
y 6 - x2 - 4x + function (Hint: Graph y = x 2 and y = 0 x 0

together. Can you transform one of the graphs

slightly to increase the number of intersections?)

5. e y 7 - 3x2 - 6x + 1
y 6 - 2x2 - 3x + 5

Lesson 4-9  Quadratic Systems 261

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Solve each system by graphing. Check your answers. See Problem 1.

8. e yy == 2-xx+2 +1 2x + 1 9. e y = x2 - 2x + 1 10. e y = x2 - x + 3
11. e yy == 2-x22x++31x + 1 y = 2x + 1 y = - 2x + 5

12. e y = -x2 - 3x + 2 13. e y = -x2 - 2x - 2
y = x+6 y = x-4

Solve each system by substitution. Check your answers. See Problem 2.

14. e yy == xx 2++14x + 1 15. e y = -x2 + 2x + 10 16. e y = -x2 + x - 1
17. e yy == x2x-2 3- 3x - 1 y = x+4 y = -x - 1

18. e y = x 2 - 3x - 20 19. e y = -x2 - 5x - 1
y = -x - 5 y = x+2

Solve each system. See Problem 3.

20. e yy == xx 22 ++ 25xx ++ 11 21. e y = x2 - 2x - 1 1 22. e y = -x2 - 3x - 2
23. e yy == 2-xx22--2xx--33 y = -x2 - 2x - y = x2 + 3x + 2

24. e y = - 3x 2 - x + 2 25. e y = x 2 + 2x + 1
y = x 2 + 2x + 1 y = x 2 + 2x - 1

Solve each system by graphing. See Problem 4.

26. e yy 77 xx 22 +- 12x 27. e y 7 x2 2--33xx 28. e y 6 -x2 - 3x
y 6 2x y 7 x2 - 1

B Apply 29. Think About a Plan  ​A manufacturer is making cardboard boxes by cutting out

four equal squares from the corners of the rectangular piece of cardboard and then

folding the remaining part into a box. The length of the cardboard piece is
1 in. longer than its width. The manufacturer can cut out either 3 * 3 in. squares,
or 4 * 4 in. squares. Find the dimensions of the cardboard for which the volume of
the boxes produced by both methods will be the same.

•  How can you represent the volume of the box using one variable?

•  What system of equations can you write?

•  Which method can you use to solve the system?

30. Open-Ended  C​ an you solve the system of equations shown by graphing? x = y 2 + 2y + 1
Justify your answer. Can you solve this system using another method? If so, ey = x-4
solve the system and explain why you chose that method.

Solve each system by substitution.

31. e xy =+ xy 2=-38x - 9 32. e y - 2x = x + 5 3 33. e y - 21x 2 = 1 + 3x
34. e xx2++yy--28==00 y + 1 = x2 + 5x y + 21 x2 = x
+

35. e xx2--1y==yx++34 36. e 2y = y - x2 + 1
y = x2 - 5x - 2

262 Chapter 4  Quadratic Functions and Equations

Graph the solution to each set of inequalities.

37. e yy 67 x-23-x 2x+-15 38. e y 6 3x 2 + 2xx++11 39. y 7 x2 - 5x + 4
y 7 2x 2 - ey 7 x2 + 3x + 2

40. Error Analysis  ​A classmate graphed the system of inequalities 6 y ≤ x2 - 4x + 6
and concluded that because the shaded regions do not intersect, 4
there are no solutions to the system. Describe and correct the error. 2

e yy Ú… xx 22 -- 44xx ++ 26 y ≥ x2 - 4x + 2

–1 2 4 6

Solve each system.

41. e yy == 3xx-2 1- 2x - 1 42. e y = -x2 + x - 5 43. y = x2 - 3x - 2
44. e yy == 21-x42x++41x212+ 4 y = x-5 ey = 4x + 28

45. y = - 3 x 2 - 4x 46. y = - 1 x 2 + x + 1
y = 4 y = 4
e 3x + 8 e x-4

47. Business  ​A company’s weekly revenue R is given by the formula R = -p2 + 30p,
where p is the price of the company’s product. The company is considering hiring a
distributor, which will cost the company 4p + 25 per week.

a. Use a system of equations to find the values of the price p for which the product

will still remain profitable if they hire this distributor.
b. Which value of p will maximize the profit after including the distributor cost?

Solve each system. 49. e y = - 7x 2 - 9x + 6
48. e yy == 5-x52x++93x + 4 y = 1
50. e yy == x-2x++32x + 6 2 x + 11

51. e y = - 4x 2 + 7x + 1
y = 3x + 2

52. Reasoning  S​ ketch the graphs of y = 2x 2 + 4x - 5 and y = x 2 + 2x - 3. Change

these equations into inequalities so the system has solutions that comprise:

a. two non-overlapping regions b. one bounded region

Solve the systems by graphing. For each system indicate one point in the
solution set.

53. e yy 76 3xx22--13 54. e y 7 x2 2 + 1 55. e y 7 (x - 3)2 + 4 5
y 6 -x y 6 - (x - 3)2 +

C Challenge 56. Find a value of a for which the line y = x + a separates the parabolas y = x2 - 3x + 2
and y = -x2 + 8x - 15.

Lesson 4-9  Quadratic Systems 263

Determine whether the following systems always, sometimes, or never have
solutions. (Assume that different letters refer to unequal constants.) Explain.

57. e yy == xx 22 ++ cd 58. y = ax 2 + c
ey = bx 2 + c

59. e yy == ((xx ++ ba))22 60. y = a(x + m)2 + c
ey = b(x + n)2 + d

61. Find the side of the square with vertical and horizontal sides inscribed in the region
y … - x2 + 1.
representing the solution of the system ey Ú x2 - 1

Standardized Test Prep

S AT/ACT 62. How many solutions does the system y = - 1 x 2- 2x
have? e y = 4 3
x2 + 4

0 1 2 3

63. Which expression is equivalent to ( -3 + 2i)(2 - 3i)?

13i 12 12 + 13i -12

64. Which expression is equivalent to (2 - 7i) , (2i)3?

S hort 78 - 1 i 14 - 7 i 7 + 1 i 41 + 7 i
Response 4 8 8 4 8

65. Solve the equation -3x 2 + 5x + 4 = 0. Show your work.

Mixed Review

Find the sum or difference. 67. (3 + 4i) - ( -4 - 3i) See Lesson 4-8.
66. (1 - i) + ( -5 + 4i) 68. (1 + i) + (2 + 2i)

Solve each equation using the Quadratic Formula. See Lesson 4-7.
71. 25x 2 - 30x + 9 = 0
69. 2m2 + 5m + 3 = 0 70. p2 - 4p + 3 = 0

Rewrite each equation in vertex form. See Lesson 4-6.
74. y = 2n2 - 8n - 3
72. y = -k 2 + 4k + 6 73. y = x 2 + 6x + 1

Get Ready!  To prepare for Lesson 5-1, do Exercises 75–77.

Simplify by combining like terms. See Lesson 1-3.

75. 3q + 9q - q 76. - 2ab2 + 2a2b + 3ab2 77. - 4y 2 + 2y + 3y 2

264 Chapter 4  Quadratic Functions and Equations

Concept Byte Powers of CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Complex Numbers
For Use With Lesson 4-9 Extendss MN-ACFNS.9A1.2 .NU-sCeNth.1e.2re  lUastieotnhei2re=la–ti1onani2d=th–e1
caonmd mthuetcaotimvem, uatsastoivciea,taivseso, acinadtivdei,starnibdudtivsteripbruotpiveerties to
Extension apdrodp,esrutibetsratocta, dand,ds mubutlrtaipctly, a.n.d. multiply . . .

MP 5

You can use the rules for multiplying complex numbers to find powers of complex
numbers.

Example 1 imaginary axis

Compute and graph (2i)n, for n = 0, 1, 2, and 3. ؊4 ؉ 0i 0 ؉ 2i axis
real
n (2i)n
0 (2i)0 ϭ 1 Ϫ4 O 1 ؉ 0i 8
1 (2i)1 ϭ 2i
2 (2i)2 ϭ 4i2 ϭ 4(Ϫ1) ϭ Ϫ4 Ϫ4i
3 (2i)3 ϭ 8i3 ϭ 8(i2 i) ϭ 8(Ϫ1 i) ϭ Ϫ8i
Ϫ8i 0 ؊ 8i

Example 2 ​ imaginary axis

Compute and graph (2 - 3i)n, for n = 0, 1, 2, and 3. 10i real axis
1 ؉ 0i
n (2 ؊ 3i)n Ϫ50 Ϫ40 Ϫ30 Ϫ20 Ϫ10 O
؊46 ؊ 9i ؊5 ؊ 12iϪ10i 2 ؊ 3i
0 (2 Ϫ 3i)0 ϭ 1
1 (2 Ϫ 3i)1 ϭ 2 Ϫ 3i
2 (2 Ϫ 3i)2 ϭ 4 Ϫ 6i Ϫ 6i + 9i2 ϭ 4 Ϫ 12i ϩ 9(Ϫ1) ϭ Ϫ5 Ϫ 12i
3 (2 Ϫ 3i)3 ϭ Ϫ10 Ϫ 24i ϩ 15i ϩ 36i2 ϭ Ϫ10 Ϫ 9i ϩ 36(Ϫ1) ϭ Ϫ46 Ϫ 9i

Exercises 265

1. Based on the graph in Example 1, predict the location of (2i)5.
2. Compute and graph ( -3i)n for n = 0, 1, 2, and 3.
3. a. Connect the points from the graph in Example 1 with a smooth curve. Estimate (2i)21.
b. Use a graphing calculator to compute (2i)12. Does it fall on the curve? Was it

close to your estimate?
4. Use a graphing calculator to find values of (2 - 3i)n for n = 0.5, 1.5, and 2.5.

Copy the graph and add these points.
5. Compute and graph (3 - 4i)n for n = 0, 1, 2, and 3.

Concept Byte  Powers of Complex Numbers

4 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in
problems, Lessons 4-2, 4-3, and 4-5. Use the work you did to complete the following.
you will pull
together many 1. Solve the problem in the Task Description on page 193 by finding all possible
concepts and prices Victor can charge for a bag of chips in order to make a profit, and
skills related determining the price he should charge to maximize his profit. Show all your
to solving work and explain each step of your solution.
quadratic
equations. Be 2. Reflect  Choose one of the Mathematical Practices below and explain how you
sure to show applied it in your work on the Performance Task.
your work and
justify your MP 1: Make sense of problems and persevere in solving them.
reasoning.
MP 2: Reason abstractly and quantitatively.

MP 4: Model with mathematics.

On Your Own

At his sandwich shop, Victor charges customers $5.00 for a sandwich, and sells an
average of 300 sandwiches per week. He wonders if lowering his selling price will
increase his sandwich sales. One week, Victor offers sandwiches at $3.00 instead
of $5.00. That week, he sells 500 sandwiches.

Victor’s cost per sandwich is $1.50. He assumes a linear relationship between the
selling price per sandwich and the number of sandwiches sold per week.

a. Determine the greatest price Victor can charge and still make a profit.

b. Determine Victor’s maximum weekly profit from sandwich sales and the
selling price that gives him that profit.

266 Chapter 4  Pull It All Together

4 Chapter Review

Connecting and Answering the Essential Questions

1 Equivalence The Different Forms of a Quadratic Modeling With Quadratics
Vertex form of a quadratic
function shows the vertex Function  (Lessons 4-1 and 4-2) (Lessons 4-3 and 4-9)
of the parabola. Standard
form is “calculator ready.” y = 2(x - 1)2 + 3 has vertex (1, 3) and opens y = - 16x2 + 12x + 4 8y
Both forms give additional upward (2 7 0). can model the height 6
information. y = - 2x2 + 4x + 1 has vertex with y in feet reached by 4
d-o2w( -4nw2)a=rd1( a-n2d x-coordinate the coin tossed by the 2x
2 Function opens referee before the game. 0 12
Any quadratic function 6 0). x represents time in
is possibly a stretch or seconds.
compression, a reflection, Each has axis of symmetry x = 1.
and a translation of y = x2. Each is a stretch of y = x2 by the factor 2.

3 Solving Equations Helpful Aids for Solving Quadratic Solving Quadratic Equations
and Inequalities
The real solutions of a Equations  (Lessons 4-4, 4-6, 4-8) (Lessons 4-5, 4-7)
quadratic equation show the
zeros of the related quadratic Factor a quadratic: -16x2 + 12x + 4 - 16x2 + 12x + 4 = 0 S
function and the x-intercepts -4(4x + 1)(x - 1) = 0 S
of its graph. = -4(4x + 1)(x - 1) 1 S
Complete the square: x2 + 4x + 1 x = - 4 or x = 1.
( ) ( )= x2 + 4x + 2+1 42
4 - 2 - 2x2 + 4x + 1 = 0
2 242
= (x + 2)2 - 3 -4 { 2( - - 4( - 2)(1)
x = 2) S
Complex numbers: x2 + 1 = (x + i )(x - i)
where i = 1-1. x = 1 + 16 or x = 1 - 126.
2

Chapter Vocabulary • factoring (p. 216) • pure imaginary number (p. 249)
• greatest common factor (p. 218) • quadratic formula (p. 240)
• absolute value of a complex number • imaginary number (p. 249) • quadratic function (p. 194)
(p. 249) • imaginary unit (p. 248) • standard form (p. 202)
• maximum value (p. 195) • vertex form (p. 194)
• axis of symmetry (p. 194) • minimum value (p. 195) • vertex of the parabola (p. 194)
• completing the square (p. 235) • parabola (p. 194) • zero of a function (p. 226)
• complex conjugate (p. 251) • perfect square trinomial (p. 219) • zero product property (p. 226)
• complex number (p. 249)
• complex number plane (p. 249)
• difference of two squares (p. 220)
• discriminant (p. 242)

Choose the correct term to complete each sentence.

1. To solve an equation by factoring, the equation should first be written in (standard
form/vertex form).

2. The value of b2 - 4ac for the equation ax2 + bx + c = 0 is called the
(discriminant/difference of two squares).

3. The number a + bi, where b = 0, is an example of a(n)(imaginary/complex) number.

Chapter 4  Chapter Review 267

4-1  Quadratic Functions and Transformations

Quick Review Exercises

You can write every quadratic function in the form Identify the vertex, axis of symmetry, maximum or
f (x) = ax2 + bx + c, where a ≠ 0. A parabola is the minimum, and domain and range of each function.
graph of a quadratic function. Every parabola has a vertex
4. f (x) = 4(x + 2)2 - 6
and an axis of symmetry. Shown below is the graph of the 5. f (x) = -(x - 3)2 + 2
quadratic parent function f (x) = x2. 6. f (x) = 10(x - 1)2 + 5
7. f (x) = 2(x + 9)2 - 4
Vertex (0, 0) 4 f(x) Axis of Symmetry Graph each function. Describe each transformation of the
Ϫ2 x=0 parent function f (x) = x2.
2 8. f (x) = x2 + 4
x 9. f (x) = (x - 9)2 + 2
10. f (x) = 12(x + 1)2 - 5
O2 Write the equation of each parabola in vertex form.
11. y
The vertex form of a quadratic function is
f (x) = a(x - h)2 + k, where a ≠ 0. The vertex of the 10
parabola formed by a quadratic function is (h, k).
8
If a 7 0, k is the minimum value of the function.
If a 6 0, k is the maximum value of the function. The axis 6
of symmetry is given by x = h.
4
Example
2
What is the vertex, axis of symmetry, maximum or x

minimum, and domain and range of the function O 246
f (x) = 5(x - 7)2 + 2?

a = 5, h = 7, k = 2 Identify a, h, and k.

vertex: (7, 2) Find the vertex: (h, k).

axis of symmetry: x = 7 The axis of symmetry is at x = h.

k = 2 is a minimum Since a 7 0, k is a minimum.

domain: all real numbers There are no restrictions on x. 12. y
6
range: y Ú 2. Since the minimum is 2, y Ú 2.

8y 4
6
4 Ϫ8 Ϫ6 Ϫ4 Ϫ2 2
2
x
O2 O2

Vertex x Ϫ4
46 8

268 Chapter 4  Chapter Review

4-2  Standard Form of a Quadratic Function

Quick Review Exercises

The standard form of a quadratic function is Graph each function.
f (x) = ax2 + bx + c, where a ≠ 0. When a 7 0, the
parabola opens up. When a 6 0, the parabola opens down. 13. f (x) = x2 + 6x + 5 14. f (x) = x2 - 7x - 18

The axis of symmetry is the line x = -2ba. The vertex is 15. f (x) = x2 - 7x + 12 16. f (x) = x2 - 9

1 - 2ba, f 1 - b 2 2, and the y-intercept is (0, c). Write each function in vertex form.
2a

17. f (x) = 4x2 - 8x + 2 18. f (x) = x2 - 8x + 12

Example 19. f (x) = 8x2 + 8x - 12 20. f (x) = -2x2 - 6x + 10

What are the vertex, the axis of symmetry and y-intercept of 21. Physics  T​ he equation h = -16t2 + 32t + 9 gives
the graph of the function f (x) = x2 - 6x + 8? the height of a ball, h, in feet above the ground,

( )axis of symmetry: x = - -6 =3 at t seconds after the ball is thrown upward. How
2(1)
many seconds after the ball is thrown will it reach its
vertex: (3, -1)
maximum height? What is its maximum height?
y-intercept: (0, 8)

4-3  Modeling With Quadratic Functions

Quick Review Exercises

You can use quadratic functions to model real world data. Find the equation of the parabola that passes through
You can find a quadratic function to model data that passes each set of points.
through any three non-collinear points given that no two of
the points lie on a vertical line. 22. (0, 5), (2, -3), ( -1, 12)
23. (2, 0), (3, -2), (1, -2)
Example 24. (4, 10), (0, -18), ( -2, -20)
25. (0, -7), (7, -14), ( -3, -19)
Find the equation of the parabola that passes through the
points ( -2, 8), (0, -2), and (1, 2).

y = ax2 + bx + c Uquseadtrhaetisctafunndcatridonf.orm of a 26. Track and Field  The table Distance Height
shows the height of a javelin (m) (m)
8 = a( - 2)2 + b( - 2) + c S ubstitute the (x, y) values to as it is thrown and travels 5 2
• - 2 = a(0)2 + b(0) + c write a system of equations. across a horizontal distance. 18 5
Use your calculator to find a 33 8
2 = a(1)2 + b(1) + c quadratic model to represent 55 6
the path of the javelin. 68 4
4a - 2b + c = 8 74 3
• c = -2

a+b+c=2

a = 3, b = 1, c = -2 S S oulbvsetitthuetesyas,tbe,manodf equations.
y = 3x2 + x - 2 c to find
the quadratic function.

Chapter 4  Chapter Review 269

4-4  Factoring Quadratic Expressions

Quick Review Exercises

To factor an expression of the form ax2 + bx + c, when Factor each expression.
a ≠ 1, you find numbers that have the product ac and sum b.
You can also factor an expression using the FOIL method in 27. x2 - 8x + 12 28. 3x2 + 11x - 20

reverse or by finding the greatest common factor (GCF). 29. - 4x2 + 14x - 6 30. x2 + 14x + 40

Example Factor each perfect square trinomial.

Factor the expression 5x2 + 13x + 6. 31. x2 - 14x + 49 32. 9x2 + 30x + 25

ac = (5)(6) = 30 Find ac. Factor each difference of two squares.

# # # #30 = 1 30 = 2 15 = 3 10 = 5 6 Foifnadct.he factors 33. 36x2 - 16 34. 25x2 - 4

b = 13 = 3 + 10 Find two factors Find the GCF of each expression. Then factor each
that sum to b. expression.

5x2 + 10x + 3x + 6 Rewrite bx. 35. 6x2 - 24x 36. - 14x2 - 49

5x(x + 2) + 3(x + 2) Find the common
factors.

(5x + 3)(x + 2) Rewrite using
the Distributive
Property.

4-5  Solving Quadratic Equations

Quick Review Exercises

The zeros of a quadratic function are the solutions of Solve each equation by factoring.
the related quadratic equation. You can find the zeros
from a table or from the x-intercepts of the parabola that 37. x2 = 4x + 12 38. 2x2 - 3x - 14 = 0
is the graph of the function. You can also find them by
factoring the standard form of a quadratic equation, 39. x2 + 2x = 8 40. x2 + 7x = 18
ax2 + bx + c = 0, and using the Zero-Product Property.
Solve each equation by graphing.

41. 5x2 + 8x - 13 = 0 42. 9 - 4x = 2x2

Example 43. x2 - x = 1 44. x2 - 2x - 4 = 0

Solve 2x2 + 6x = 8 by factoring. Solve each equation by using a table.
2x2 + 6x - 8 = 0 Rsteawndriaterdthfoeremq.uation in
45. x2 - 6x + 8 = 0 46. 9x - 14 = 3x2

2(x2 + 3x - 4) = 0 Factor out the GCF, 2. 47. x2 - 5x + 2 = 0 48. 2x2 - 12x = - 16

2(x + 4)(x - 1) = 0 Factor the quadratic expression.

2(x + 4) = 0 or x - 1 = 0 Use the Zero-Product Property.

x = - 4 or x = 1 Solve.

270 Chapter 4  Chapter Review

4-6  Completing the Square

Quick Review Exercises

If you cannot solve a quadratic equation by factoring, you Solve each equation by finding square roots.
can use completing the square. You write one side as a
perfect square trinomial and then take square roots. You 49. 4x2 = 16 50. 4x2 - 20 = 0
can also convert a quadratic function from standard form to
vertex form by completing the square. 51. 5x2 - 45 = 0 52. 3x2 = 36

What values complete each square?

Example 53. x2 - 6x 54. x2 + 3x

Solve x2 + 6x - 7 = 0 by completing the square. Solve each equation by completing the square.

x2 + 6x = 7 iRse bwyriittesetlhf.e equation so the constant 55. x2 + 8x + 6 = 0 56. x2 - 10x = 13

( ) ( ) ( )b2 2 57. 9x2 + 6x + 1 = 4 58. x2 - 2x + 4 = 0

2 6 b 2.
= 2 = 32 = 9 Find 2 59. x2 + 3x = - 25 60. 4x2 - x - 3 = 0

( )x2 + 6x + 9 = 7 + 9 Add b 2 to each side.
2
(x + 3)2 = 16
Factor and simplify.

x + 3 = { 4 eTaakceh the square root of
side.

x = 1 or x = -7 Solve for x.

4-7  The Quadratic Formula

Quick Review Exercises

You can solve a quadratic equation in the form Solve each equation using the quadratic formula.
ax2 + bx + c = 0 by using the Quadratic Formula,
61. 3x2 + 5x = 8 62. x2 = 6x - 9
x = -b { 2b2 - 4ac. The discriminant of a quadratic
2a 63. x(x - 3) = 4 64. 5x2 - 7x - 3 = 0
equation in standard form is the value of the expression
b2 - 4ac. You can use it to find the quantity and type of
Determine the discriminant of each equation. How many
solutions of a quadratic equation. real solutions does each equation have?

Example 65. 4x2 - 2x = 10 66. x2 - 5x + 7 = 0

Use the Quadratic Formula to solve 2x2 - 6x = -3. 67. 3x2 + 3 = 6x 68. 7 - 3x = 8x2

2x2 - 6x + 3 = 0 Write the equation in 69. Gardening  Margaret is planning a rectangular
standard form. garden. Its length is 4 ft less than twice its width.
Its area is 170 ft2. What are the dimensions of the
a = 2, b = - 6, c = 3 Identify a, b, and c. garden?

x = - ( - 6) { 2( - 6)2 - 4(2)(3) tS huebsqtuitaudtreaati,cbf,oarmndulcai.nto
2(2)

x = 6 { 112 = 3 { 13 Simplify.
4 2

Chapter 4  Chapter Review 271

4-8  Complex Numbers

Quick Review Exercises

A complex number is written in the form a + bi, Simplify each expression using the imaginary unit i.
where a and b are real numbers, and i is equal to 1-1.
If b = 0, a + bi is a real number. If b ≠ 0, a + bi is an 70. 1-24 71. 1-2 - 3
imaginary number. You can use the Quadratic Formula
or completing the square to find the imaginary solutions of 72. (4 + 1-25)( 2 -100) 73. 21-24 + 6
quadratic equations.
Simplify each expression.

74. (9 + 7i) - (6 - 2i) 75. (3 + 11i) + (10 + 9i)

Example 76. (1 - 9i)(3 + 2i) 77. (3i)2 - 3(1 + 5i)

Use the Quadratic Formula to solve 3x2 - 4x + 2 = 0. 78. 4 - 6i 79. 12 - 3i
2i + 5i
- ( - 4) { 2( - 4)2 - 4(3)(2) qE nutaedrraat,ibc ,foanrmd uclain. to the
x = 2(3) Solve each equation.

x = 4 { 116 - 24 = 4 { 1 - 8 Simplify. 80. x2 + 9 = 0 81. 5x2 - 2x + 1 = 0
6 6
82. - x2 + 4x = 10 83. 7x2 + 8x = - 6

x = 2 { 12 i Write the solutions.
3 3

4-9  Quadratic Systems

Quick Review Exercises

A system of quadratic equations can be solved by Solve each system by substitution.
substitution or by graphing. You can use these methods to
solve a linear–quadratic system or a quadratic–quadratic 84. y = x2 - 7x - 6
system. Use graphing to solve a quadratic system of ey = 8 - 2x
inequalities.
85. y = - x2 - 2x +8
ey = x2 - 8x - 12

Example y = x22x2++5x2x--610. Solve each system by graphing.
y =
Use substitution to solve e y = - x2 - 10x + 12
ey = x2 - 6x - 18
2x2 + 2x - 10 = x2 + 5x - 6 Substitute for y. 86.

x2 - 3x - 4 = 0 Rewrite in standard form. 87. y = x2 - x - 18
ey = 2x + 3
(x + 1)(x - 4) = 0 Factor.

x = -1 or x = 4 Solve for x. Solve each system of inequalities.

y = ( - 1)2 + 5( - 1) - 6 = - 10 sSoulbvset iftourtey.for x then 88. y 6 x+4 + 2
ey Ú x2 + 2x

y = (4)2 + 5(4) - 6 = 30 89. y 7 3x2 - 10x - 8
ey 6 x2 - 5x + 4
( - 1, - 10) and (4, 30) Wordrieteresdo lpuatiirosn. s as

272 Chapter 4  Chapter Review

4 Chapter Test M athX

OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Solve the following systems of equations.
Sketch a graph of the quadratic function with the given
vertex and through the given point. Then write the 17. y = 3x 2 - x + 1
equation of the parabola in vertex form and describe ey = 3x 2 + x - 1
how the function was transformed from the parent
function y = x2. 18. y = -x2 + 2x - 3
ey = 4x - 3
1. vertex (0, 0), point ( -3, 3)
Solve the following systems of inequalities.
2. vertex (1, 5), point (2, 1)
19. y 7 2x 2 + 5x + 1
ey 6 - 2x 2 - 5x - 1
Graph each quadratic function. Identify the axis of
symmetry, the vertex, and the domain and the range of 20. y 6 x2 - x + 2
each function. ey 7 x2 - 1

3. y = x2 - 7 Evaluate the discriminant of each equation. How many
4. y = x2 + 2x + 6 real and imaginary solutions does each have?

5. y = -x2 + 5x - 3 21. x 2 + 6 x - 7 = 0
22. 3x 2 - x + 3 = 0
Simplify each expression. 23. - 4x 2 - 4 x + 1 = 0

6. 1 - 16

7. 41-9 - 2 Do you UNDERSTAND?

8. (2 + 3i)(8 - 5i) 24. Writing  ​Compare graphing a number on the
complex plane to graphing a point on the coordinate
9. ( -3 + 2i) - (6 + i) plane. How are they similar? How are they different?

10. 4+ 2i 25. Open-Ended  Sketch the graph of a quadratic
2- i function f (x) = ax2 + bx + c that has no real zeros.
How does this relate to the solutions of the related
Factor each expression completely. equation ax2 + bx + c = 0?
11. 2y2 - 8y
12. 3x2 + 8x - 3 STEM 26. Physics  A​ model for the path of a toy rocket is given
13. 9w2 - 30w + 25 by h = 68t - 4.9t2, where h is the altitude in meters
and t is the time in seconds. Explain how to find both
Solve each quadratic equation. the maximum altitude of the rocket and how long it
14. x 2 - 25 = 0 takes to reach that altitude.
15. x 2 - 2x + 3 = 0
16. x 2 - 8x = - 6 27. How many solutions are possible for:
a. a system of two quadratic equations?
b. a system of two quadratic inequalities?
Explain your answers.

Chapter 4  Chapter Test 273

4 Common Core Cumulative ASSESSMENT
Standards Review

Some questions on tests require Roy has a 400 foot roll of wire. He wants to TIP 2
that you model a word problem use it to fence in a rectangular area. What is
with a quadratic function. the maximum area of the enclosed space? Use the information from
the problem. The perimeter
TIP 1 20,000 square feet is 400, so 2(O + w) = 400.
Solve for w : w = 200 − O.
To identify the function, 10,000 square feet
use what you already Think It Through
know. You know that the 200 square feet
perimeter of a rectangle Substitute for w in the
100 square feet
#is 2(O + w) and the area #area formula:
A = f (/) = / (200 - / )
is O w. = - / 2 + 200/.

The maximum value is the

( )y-coordinate of the vertex,b
f - 2a = f (100) = 10,000.

So, the maximum area Roy can

enclose is 10,000 square feet.

The correct answer is B.

LVVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand the Read each question. Then write the letter of the correct
meanings of mathematical terms. Match each term answer on your paper.
with its mathematical meaning.
1. Which equation is equivalent to
A . axis of I.  value of b2 - 4ac for the x2 + 24x + 100 = - 46?
symmetry equation ax 2 + bx + c = 0
(x + 12)2 = - 2
B . discriminant II.  If ab = 0, then a = 0 or b = 0.
(x - 12)2 = - 2
C. imaginary III.  line that divides a parabola
(x - 12)2 = 2
number into two parts that are mirror
(x + 12)2 = 2
images
D . Zero-Product 2. What is the solution of the following system of
equations?
Property IV.  a + bi, a and b are real
E. parabola numbers and b ≠ 0 23xx - y = 4
+ y = 1
F. perfect sq uare V.  square of a binomial

trinomial VI.  process of finding the last ( -1, 2)
(1, -2)
G . cthoemspqlueatirne g tterirnmomtoiaml ake a perfect square (2, 1)

VII.  graph of a quadratic ( -2, 1)

function

274 Chapter 4  Common Core Cumulative Standards Review

3. What are the factors of the quadratic function graphed 8. What are the domain and range of the function

below? y graphed below? 4y
(x + 3) and (x + 2) 2
Domain: All real
x and (x - 6) 6 numbers

x and (x + 6) 4 Range:

(x - 3) and (x + 2) All real numbers … 3 x

2 Domain: All real Ϫ4 Ϫ2 O 2

numbers Ϫ2

Ϫ2 O 2 4 6x Range:
Ϫ2
All real numbers Ú 3

4. Which equation has -1 { i as its solution? Domain: All real
x 2 - 2x - 2 = 0 numbers between -5 and -1
2x 2 - 2x - 1 = 0
2x 2 + 2x + 1 = 0 Range: All real numbers … 3
x 2 + 2x + 2 = 0
Domain: All real numbers between -5 and -1
Range: All real numbers Ú 3

5. What are the solutions of 0 3x - 5 0 = 2? 9. The formula for the total surface area of a regular right

pentagonal prism is A = ap + pH. Solve this equation
for p.

7 p = a + H p = A - a
3 A H
x = -1 and x =

7 p = a A H p = H - a
3 + A
x = 1 and x =

x = 1 and x = 1 1 0. What is the solution of e -y = 3x - 21?
5 2y = -x -

x = -1 and x = 1 x = 20, y = -11 x = -20, y = 11
5

6. What is the transformation of the graph of x = 45, y = - 7 x = - 54, y = 7
y = (x + 3)2 - 2 from its parent function y = x 2? 5 5

3 units left and 2 units down 11. Which system of inequalities is graphed below?

3 units right and 2 units up y … -1 2 4y
ey + xÚ
6 units right and 2 units up
y Ú -1 2 2
2 units left and 3 units down ey + x… x

7. What is the axis of symmetry for the graph of the y 6 1 Ϫ4 Ϫ2 O 24
quadratic equation y = -3x 2 - 12 + 12x? y + x Ϫ2
e 7 -2
x = -2
x = 2 e y 7 1 Ϫ4
x = 12 y + x
x = -12 6 -2

1 2. What is the vertex of y = -2 0 x + 4 0 - 5?

( -2, -5) (4, -5)

( -4, -5) (2, -5)

Chapter 4  Common Core Cumulative Standards Review 275

Selected Response 26. What are the solutions of the system? Solve by
graphing.
1 3. What is the sum of the solutions of the equation
1.5x 2 - 2.5x - 1.5 = 0? Round to the nearest yy = x2 - x - 2
hundredth. = -x + 2

14. What is the value of y in the system of equations? 2 7. A swimmer swam 1000 meters downstream in
15 minutes and swam back in 30 minutes against the
yx + y = 10 current. What was the rate of the swimmer in still
= 2x + 1 water? How fast was the current?

15. Z eroy and Darius shop at the mall during the special 28. Claudia has a rectangular flowerbed. She decided
sale extravaganza. Zeroy spends $120 on 3 pairs of that the original width w, in feet, was too small, so she
pants and 4 shirts. Darius buys 2 pairs of pants and increased the width by 3 feet. She also changed the
3 shirts and spends $85. What is the price of one shirt? length to be 1 foot less than twice the original width.
What is an expression that represents the area of the
16. W hat is the sum of the solutions of new flower bed?

0 5x - 4 0 = 8 - x? 29. Explain how you would graph y + 4 6 2 0 x - 3 0 on a

17. Suppose y varies directly with x, and y = -4 when coordinate grid.
x = 5. What is the constant of variation?

1 8. Molly is making a punch for the school picnic. Extended Response
3
The recipe calls for 4 quart of lemonade, 3 cups of 3 0. A hat company is designing a one-size-fits-all hat with
a strap in the back that makes the hat smaller or larger.
cranberry juice, 4 cans of orange juice concentrate, and Head sizes normally range from 51 to 64 centimeters.
What absolute value inequality models the different
5 cups of water. If Molly uses 125 quarts of lemonade, sizes of the hat? Graph the solution.
how many cups of cranberry juice will she need?
3 1. Robby decided to earn extra money by making and
1 9. What is the value of 3x2 - 5x + 7 when x = 52? selling brownies and cookies. He had space in his
Express the answer as a decimal. oven to make at most 80 brownies and cookies. Each
brownie cost $.10 to make and each cookie cost $.05
2 0. What is the real part of (11 + 10i)(2 + 3i )? to make. He had $6 to spend on ingredients.

21. H ow many imaginary roots does 2x2 + 3x - 5 = 0 a. Write a system of inequalities to represent the
have? situation.

22. What is the product of (4 + 3i )(4 - 3i )? b. Graph the system, choose one point in the
feasible region, and explain what the point means
23. What is the greatest integer solution of in terms of the problem.

0 2x + 3 0 - 4 … 0? c. If Robby makes a profit of $.25 on each brownie
and $.20 on each cookie, how many of each
2 4. What is the coefficient of the x-term of the dessert should he make to maximize his profit?
factorization of 25x2 + 20x + 4?

2 5. A piggy bank contains $2.40 in nickels and dimes. If
there are 33 coins in all, how many nickels are there?

276 Chapter 4  Common Core Cumulative Standards Review