chapter 4 prerequisite skills - mrs. lesieur's online...

Copyright © 2011, McGraw-Hill Ryerson Limited, ISBN: 978-0-07-073883-6

Name: ___________________________________________ Date: _____________________________

BLM 4–2

Chapter 4 Prerequisite Skills 1. Multiply and combine like terms.

a) 3(2x – 7) – 4(x – 1) b) 5x(3x – 2) c) (4x – 3)(2x + 5) d) (5x – 4)2

2. Factor fully. a) 3xy – 8x2y b) 3p – 9p2 c) x2 – 13x + 12 d) 4a2 – 9y2

e) 8r2 + 20r + 8 f) 2x2 – 0.08y2

3. Solve for x. Check your answer for part a). a) 7x – 3 = 2x – 5 b) 19 – 2(x + 3) = 1 c) (4x + 3)(x – 1) = (2x – 1)(2x + 1)

4. Use the graph to help determine each of the following:

a) coordinates of the vertex b) equation of the axis of symmetry c) range of the function d) y-intercept e) x-intercepts

5. A quadratic function is represented by g(x) = 2x2 – 6x + 3. a) What is the value of the function

when x = −1?

b) What is 32

g ⎛ ⎞⎜ ⎟⎝ ⎠

?

6. For the equation y = –2(x – 3)2 + 5, explain how you can identify each of the following without graphing: a) the equation of the axis of symmetry b) coordinates of the vertex c) whether the function has a maximum or

minimum value d) the nature of the x-intercepts e) value of the y-intercept

7. Sketch each parabola. Label the vertex and axis of symmetry. a) y = (x + 3)2 – 4 b) y = –2(x – 1)2 – 5

8. Convert each equation to the form y = ax2 + bx + c. What are the values of a, b, and c? a) y = –3(x – 1)2 + 2

b) 232

( 4) 5y x= + −

9. Convert each equation to the form y = a(x – p)2 + q by completing the square. Determine the value for a, p, and q in each case. a) y = x2 – 10x + 31 b) y = 6x2 + 24x + 17 c) y = –4x2 + 20x – 3

10. A business models its costs with the function C(n) = 30n2 – 720n + 6000, where C(n) is the cost, in dollars, of producing n items. Determine the number of items that must be produced to create the minimum cost for the business.


BLM 4–3

Chapter 4 Warm-Up Section 4.1 Warm-Up 1. Quadratic functions are represented by

f (x) = 2x2 – x + 3 and g(x) = 2(x –3)2 + 7. a) What is the value of f (−4)? b) Determine g(2). c) What is g(5) − f (0)?

2. a) Complete the table of values for the quadratic function f (x) = –3x2 + 6x – 1. Then, answer the questions below.

x −2 −1 0 1 2 3 4 f (x)

b) What is the equation for the axis of symmetry?

c) What is the maximum value of this function?

d) How could symmetry help you complete the table of values if you knew the equation for the axis of symmetry?

3. Sketch a graph of each quadratic equation. Determine the coordinates of the vertex and the number of real x-intercepts for each graph. a) y = (x – 3)2 + 2 b) y = –2(x + 1)2 + 5 c) y = 3(x + 2)2

4. Use the graph to help answer the following questions:

a) What is the equation for the axis

of symmetry? b) What are the coordinates of the vertex? c) Determine the x-intercepts for the curve. d) If the equation for the function is written in

the form y = ax2 + bx + c, what can you say for certain about the coefficient a?

5. Sketch a parabola that satisfies each set of information. a) vertex at (−1, 2) and y-intercept of −3 b) Equation of axis of symmetry is x = 4.

Vertex is below the x-axis and the parabola opens upward.

c) vertex at (0, 2) and has no x-intercepts

Section 4.2 Warm-Up 1. Multiply and combine like terms.

a) (x – 4)(x + 3) b) 5x(2x – 5) – 3x(1 – x) c) (3x – 7)(2x + 1) d) (2x – 3)2 e) (2x – 7)(2x + 7)

2. The dimensions of a rectangle are given as x + 5 and 2x − 3, in centimetres. Determine a simplified expression for a) the perimeter of the rectangle b) the area of the rectangle

3. Solve each equation for the variable x. a) 5(x – 3) – 2(4x – 1) = 5 b) (2x – 5)(x + 3) = 2(x – 3)(x + 3)

4. Factor each trinomial. a) x2 + 4x – 21 b) x2 + 7x + 10 c) 2x2 – 7x + 6 d) 4x2 + 11x – 3

5. Factor each expression fully. a) –3x2 + 9xy – 6y2 b) –4x2 + 49 c) 2x2 – 12x + 18 d) 3x3 – 3x2 + 27x e) x4 – 16


BLM 4–3 (continued)

Section 4.3 Warm-Up 1. Answer the following questions using your

knowledge of square roots.

a) Is a decimal value of 12 considered exact or approximate?

b) What is the value of 28 correct to three decimal places?

c) How many solutions does the equation x2 = 9 have?

d) What two whole numbers does 19 fall between?

e) What is the solution for the equation 364 2 ?xx x= −

2. Solve each quadratic equation. a) x2 – 36 = 0 b) 3x2 – 12 = 0

c) 214

1 24y − =

d) (x – 3)2 = 81 e) (2p – 1)2 + 4 = 53

3. Write each equation in the form ax2 + bx + c = 0, where a, b, and c are integers and a ≠ 0. a) (2x – 5)(x – 1) = 0 b) 3(x – 4)2 = –5 c) roots are 2 and −3

d) roots are 23

and −2

4. What number must be added to each expression to create a perfect square trinomial? a) x2 – 8x b) m2 + 24m c) y2 – 3y

d) 2 23

n n−

5. Solve the equation (x + 2)2 – (x + 2) = 42 using two different algebraic methods.

Section 4.4 Warm-Up

1. Two quadratic functions are defined by f (x) = –2(x + 3)2 + 1 and g(x) = 3x2 + 6x + 1. a) Express function g in the form

g(x) = a(x – p)2 + q. b) In a graph of the functions, which

parabola’s vertex will be closest to the x-axis?

c) Which parabola has a y-intercept of 1?

d) What is the range of function f ? e) Express function f in the form

f (x) = ax2 + bx + c. f) What is the equation of the axis of

symmetry for function g? g) Sketch the graph of function f.

2. Solve each quadratic equation by completing the square. Express answers to the nearest tenth. a) x2 – 8x + 13 = 0 b) –3x2 + 4x = –5

3. Rewrite each quadratic equation in the form ax2 + bx + c = 0. Then, identify the numerical values of a, b, and c in each case.

a) 234

1 6x x= −

b) – (x – 2)2 = 5

4. Solve each quadratic equation using different methods. a) x2 + 6x – 4 = 0 b) 2x2 + x – 6 = 0 c) x2 – 8x + 15 = 0


Name: ___________________________________________ Date: _____________________________

BLM 4–4

Section 4.1 Extra Practice 1. How many x-intercepts does the graph of

each quadratic function have? a) b) c)

d)

2. What are the roots of the quadratic equations graphed in #1?

3. Solve by graphing. a) 0 = –a2 – 3a – 4 b) 12 = –3b2 – 12b c) 6c2 + 30c = 0 d) d

2 – 4 = 0

4. Determine the roots for each quadratic equation. Where integral roots cannot be found, estimate the roots to the nearest tenth. a) 0 = x2 + 2.4x – 3.85 b) z2 – 15 = 0 c) t

2 + t = –1 d) 0 = –u2 – u + 5

5. Solve by graphing. a) t

2 – 5t – 150 = 0 b) h2 – 400 = 0 c) 0 = x2 + 0.6x – 0.05 d) 5y

2 + 3y + 100 = 0


Name: ___________________________________________ Date: _____________________________


6. For what values of m would the equation x2 + 8x + m = 0 have a) one real root or two equal real roots? b) two real distinct roots? c) no real roots?

7. An object is launched at 21.5 m/s from a height of 2.4 m. The equation for the object’s height, h, measured in metres, t seconds after launch is h = –4.9t2 + 21.5t + 2.4. After how many seconds will the object hit the ground? Express your answer to the nearest tenth of a second.

8. A right triangle has one side that is 7 cm longer than its shortest side. The triangle’s hypotenuse is 8 cm longer than the shortest side. What are the dimensions of the triangle?


Name: ___________________________________________ Date: _____________________________

BLM 4–5

Section 4.2 Extra Practice 1. Factor.

a) x2 – x – 20 b) 3x2 – 30x + 63 c) – 4x2 – 12x – 8

d) 21 12 2 6x x− −

2. Factor. a) 14x2 + 3x – 5 b) 3x2 + 11x – 20 c) 4x2 + 7xy + 3y2 d) 6x2 – 17x + 12

3. Factor completely. a) 12x2 – 4xy – 8y2 b) 6x2y + 27xy + 30y c) 140x2 – 450xy + 250y2 d) 42x3 + 77x2y + 21xy2

4. Factor. a) x2 – 49y2 b) 25x2 – 9

c) 2 2254x y−

d) (x + 1)2 – (x – 7)2

5. Factor. a) (x – 1)2 – 2(x – 1) – 35 b) 6(2x + 1)2 – 7(2x + 1) – 20 c) 2(7x)2 + 2(7x) – 24

d) ( ) ( )2

2 21 12 28 6 9x x− −

6. Solve each quadratic equation by factoring. Verify your answer. a) x2 – 2x – 15 = 0 b) 2x2 + 8x = 64

c) 2 912 2 9 0x x− + =

d) 7x2 – 35 = 0

7. Solve each quadratic equation. a) 6x2 – 5x = 4 b) 7x2 = 34x + 5 c) 5x2 = 9x + 2 d) 2x2 + 9x = 18

8. Determine the real roots of each quadratic equation. a) 64x2 – 169 = 0 b) 18x2 – 98 = 0 c) 80x2 = 5 d) (x + 1)2 – 81 = 0

9. Determine the real roots to each quadratic equation by factoring. a) 6x2 + 2x – 4 = 0 b) 10x2 – 45x + 20 = 0 c) 18x2 = 3x + 3

d) 2 52 21 0− − =x x

10. Solve each quadratic equation. a) 9x2 + 6x + 1 = 0 b) 20x2 – 60x + 45 = 0

c) 2 2545 0x x+ + =

d) 1.6 – 5.6x + 4.9x2 = 0


Name: ___________________________________________ Date: _____________________________

BLM 4–6

Section 4.3 Extra Practice 1. What value of k makes each expression a

perfect square? a) x2 + 12x + k

b) x2 – 20x + k c) x2 – 7x + k

d) 2 45x x k+ +

2. Complete the square to write each quadratic equation in the form (x + a)2 = b. a) x2 + 6x + 4 = 0 b) 2x2 – 16x + 10 = 0 c) –3x2 + 15x – 2 = 0

d) 212 5 4 0x x+ − =

3. Solve each quadratic equation, to the nearest tenth. a) (x – 4)2 = 25

b) ( )21 12 4x + =

c) (x – 0.1)2 = 0.64 d) 4(x + 7)2 = 1

4. Solve each quadratic equation. Express answers as exact roots in simplest form. a) x2 + 2x – 2 = 0 b) x2 – 5x + 3 = 0 c) x2 + 0.6x – 0.16 = 0

d) 2 6 9 07 49x x− + =

5. Solve each quadratic equation by completing the square. Express answers in simplest radical form. a) 4x2 + x – 3 = 0 b) –3x2 – 6x + 1 = 0

c) 214 5 0x x+ − =

d) –0.1x2 + 0.6x – 0.5 = 0

6. Solve each quadratic equation by completing the square. Express answers to the nearest hundredth. a) –2x2 + 9x + 2 = 0 b) 3x2 – 3x – 1 = 0

c) 215 2 1 0x x+ + =

d) 6x2 + 3x – 2 = 0

7. Two numbers have a sum of 22. What are the numbers if their product is 96?

.


Name: ___________________________________________ Date: _____________________________

BLM 4–7

Section 4.4 Extra Practice 1. Use the discriminant to determine the nature

of the roots for each quadratic equation. Do not solve the equation. a) 7x2 + x – 1 = 0

b) 3x2 – 4x + 5 = 0 c) 8y2 – 8y + 2 = 0 d) 3x2 + 6 = 0

2. Without graphing, determine the number of zeros for each quadratic function. a) f (x) = 3x2 – 2x + 9 b) g(x) = 9x2 – 30x + 25 c) h(t) = – 4.9t2 – 5t + 50 d) A(x) = (x + 5)(2x – 1)

3. Use the quadratic formula to solve each quadratic equation. Express answers as exact values in simplest form. a) x2 – 10x + 23 = 0 b) 4x2 – 28x + 46 = 0 c) 9x2 – 12x = – 4 d) 10x2 – 15x = 0

4. Use the quadratic formula to solve each quadratic equation. Express answers to the nearest hundredth. a) 6x2 – 5x + 1 = 0 b) – 0.1x2 + 0.12x – 0.08 = 0 c) –3x2 + 5x + 4 = 0

d) 2 2

5 31 0x x+ − =

5. Determine the real roots of each quadratic equation. Express your answers as exact values. a) x2 + 4x – 1 = 0 b) 4x2 – 4x – 7 = 0 c) 8x2 + 20x + 11 = 0 d) x2 – 4x – 3 = 0

6. Solve each quadratic equation using any appropriate method. Express your answers as exact values. Justify your choice of method. a) x2 + 4x + 10 = 0 b) x2 + 7x = 0 c) 4x2 + 20x + 25 = 0 d) (x + 4)2 = 3 e) 6x2 + 2x – 1 = 0

7. For the quadratic equation 2x2 + kx – 2 = 0, one root is 2. a) Determine the value of k. b) What is the other root?

.


Name: ___________________________________________ Date: _____________________________

BLM 4–8

Chapter 4 Review #22 Cut out and arrange the algebraic steps and explanations in the order necessary to arrive at the quadratic formula.

Algebraic Steps Explanations

ax2 + bx = –c Take the square root of both sides.

2 2

2

2 24 4b b b cx xa aa a

+ + = −

Subtract c from both sides.

2 b cx xa a+ = −

Divide both sides by a.

2

24

2 4b b acx a a

−+ = ±

Complete the square.

2 4

2

b b acx

a

− ± −=

Factor the perfect square trinomial.

2 2

24

2 4b acbx a a

−+ =⎛ ⎞⎜ ⎟⎝ ⎠

Solve for x.


Name: ___________________________________________ Date: _____________________________

BLM 4–9

Chapter 4 Test Multiple Choice For #1 to 5, choose the best answer.

1. Consider the quadratic function f (x) = 2x2 – 8x – 5. The smallest zero of the function is A –0.55 B –5.00 C 2.00 D 4.55

2. The roots of the quadratic equation 6x2 – 16x = 0 are

A 0 B 0 or 83

C 2 or 83

D 83

3. For what value of k does the equation (2k – 1)x2 – 8x + 2 = 0 have two equal real roots?

A 12

B 292

C 72

D 92

4. Which student uses correct mathematical vocabulary to describe the solutions to a quadratic equation? A Alain: The solutions are the roots of the

quadratic function. B Beth: The solutions are the zeros of the

quadratic function. C Cody: The solutions are the x-intercepts

of the quadratic equation. D Dolores: The solutions are the

y-intercepts of the graph of the related function.

5. Which graph represents a quadratic function that has two distinct real roots? A B C D


2

2

2

2

22 315

15 6 2 015 6 2

15 6 9 2(15 3) 2

15 3

x xx x

x xx

x

x −

+ − =

+ =

+ + =

+ =

+ =

=

2( 4) 4 4(2)( 3)2(2)

4 16 244

4 84

2 22

x

x

x

x

− − ± − − −

± − +

±

±

=

=

=

=

Name: ___________________________________________ Date: _____________________________


Short Answer 6. A smokejumper is a firefighter who parachutes

into remote areas to combat forest fires. Saskatchewan’s smokejumpers, founded in 1949, were Canada’s first aerial firefighting team. The function h(t) = –16t2 + 1500 models the height, h, of a smokejumper, in feet, t seconds after jumping from 1500 ft. Suppose a parachute opens at 1000 ft. Determine algebraically how long the jumper was in free fall, to the nearest hundredth of a second.

7. Identify and correct the errors in each solution to the quadratic equations. a) 2x2 – 4x –3 = 0 b)

8. Determine the real roots of each equation

algebraically. Choose a different method for each equation, and explain why you chose that method. Express your answers as exact values in simplest form. a) x2 – 10x + 16 = 0 b) 3x2 + 19x – 14 = 0 c) x2 – 6x + 7 = 0 d) 2(x – 3)2 – 8 = 0

9. Rewrite the equation 1 2 3 15 5

x x xx x x+ − −− −

+ = as a simplified

quadratic equation equal to zero. Then, use the quadratic formula to determine the real roots of the equation.

10. For what values of k does the graph of f (x) = kx2 – 5x + k have no x-intercepts?

Extended Response 11. The length and width of a rectangle are

7 m and 5 m, respectively. When each dimension is increased by the same amount, the area is tripled. Find the dimensions of the new rectangle, to the nearest tenth of a metre.

12. Find a rational number such that the sum

of the number and its reciprocal is 136

.

13. Robin Chestnut is a two-time Canadian juggling champion. As part of his act, Robin tosses a ball into the air and lets it drop to the floor. After a ball is tossed, its height, h, in metres, after t seconds, is modelled by the equation h(t) = – 4.9t2 + 12t + 1.5. For how many seconds, to the nearest hundredth, is the ball in the air?


BLM 4–10

Chapter 4 BLM Answers BLM 4–2 Chapter 4 Prerequisite Skills 1. a) 2x – 17 b) 15x2 – 10x c) 8x2 + 14x – 15 d) 25x2 – 40x + 16 2. a) xy(3 – 8x) b) 3p(1 – 3p) c) (x – 1)(x – 12) d) (2a – 3y)(2a + 3y) e) 4(2r + 1)(r + 2) f) 2(x – 0.2y)(x + 0.2y)

3. a) 25

x −= Check:

2 25 5

29 295 5

7 3 2 5− −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− = −

=

b) x = 6 c) x = –2 4. a) (2, −9) b) x = 2 or x − 2 = 0 c) {y | y ≥ –9} d) (0, −5) or y-intercept −5 e) (−1, 0) and (5, 0)

5. a) 11 b) 32

−

6. a) Equation for the axis of symmetry is x = p, so in this case it is x = 3. b) Vertex at (p, q), so it is at (3, 5). c) Since a is negative, the parabola opens downward, and the function has a maximum value. d) Parabola opens down from (3, 5), so there are two x-intercepts. e) x = 0 at the y-intercept. Substitute to get y = −13. The y-intercept occurs at (0, −13). 7. a) b)

8. a) y = –3x2 + 6x – 1; a = –3, b = 6, c = –1 b) 23

212 19;y x x= + + a = 1.5, b = 12, c = 19

9. a) y = (x – 5)2 + 6; a = 1, p = 5, q = 6 b) y = 6(x + 2)2 – 7; a = 6, p = –2, q = –7

c) 25 5

2 24 22; = 4, , = 22y x a p q⎛ ⎞⎜ ⎟⎝ ⎠

= − − + − =

10. 12

BLM 4–3 Chapter 4 Warm-Up Section 4.1 1. a) 39 b) 9 c) 12 2. a) x −2 −1 0 1 2 3 4 f (x) −25 −10 −1 2 −1 −10 −25

b) x = 1 c) 2 d) Function values are the same on opposite sides of the axis of symmetry, so you really only need to find values for half the table. 3. a)

vertex: (3, 2); no x-intercepts b)

vertex: (–1, 5); two x-intercepts c)

vertex: (–2, 0); one x-intercept



4. a) x = 1 b) (1, −4) c) (−1, 0) and (3, 0) d) a is positive or a > 0. 5. a) Example:

b) Example:

c) Example:

Section 4.2 1. a) x2 – x – 12 b) 13x2 – 28x c) 6x2 – 11x – 7 d) 4x2 – 12x + 9 e) 4x2 – 49 2. a) 2(x + 5) + 2(2x – 3) = (6x + 4) cm b) (x + 5)(2x – 3) = (2x2 + 7x – 15) cm2 3. a) x = −6 b) x = −3 4. a) (x + 7)(x – 3) b) (x + 5)(x + 2) c) (2x – 3)(x – 2) d) (4x – 1)(x + 3) 5. a) –3(x2 – 3xy + 2y2) = –3(x – y)(x – 2y) b) 49 – 4x2 = (7 – 2x)(7 + 2x) c) 2(x2 – 6x+ 9) = 2(x – 3)(x – 3), or 2(x – 3)2 d) 3x(x2 – x + 9) e) (x2 – 4)(x2 + 4) = (x – 2)(x + 2)(x2 + 4)

Section 4.3 1. a) approximate b) 5.292 c) 2 d) 4 and 5 e) x = 0, x = 1 2. a) x = ±6 b) x = ±2 c) y = ±10 d) x = 12 or x = –6 e) p = 4 or p = –3

3. a) 2x2 – 7x + 5 = 0 b) 3x2 – 24x + 53 = 0 c) x2 + x – 6 = 0 d) 3x2 + 4x – 4 = 0

4. a) 16 b) 144 c) 94

d) 19

5. Method 1: Multiply and solve: x2 + 4x + 4 – x – 2 = 42 x2 + 3x – 40 = 0 (x – 5)(x + 8) = 0 x = 5 or x = –8

Method 2: Substitute for x + 2 and solve. Then substitute back: Let m = x + 2. m2 – m – 42 = 0 (m + 6)(m – 7) = 0 m = –6 or m = 7 Then, –6 = x + 2, so x = –8, or 7 = x + 2, so x = 5.

Section 4.4 1. a) g(x) = 3(x + 1)2 – 2 b) function f c) function g d) {y | y ≤ 1} e) f (x) = –2x2 – 12x – 17 f) x = −1 g) 2. a) x = 5.7 or x = 2.3 b) x = –0.8 or x = 2.1

3. a) 234

6 1 0,x x+ − = 34

a = , b = 6, c = –1, or

3x2 + 24x – 4 = 0, a = 3, b = 24, c = –4 b) –x2 + 4x – 9 = 0, a = –1, b = 4, c = –9 4. Example: a) complete the square: 133x = − ±

b) factor: 32

x = or x = –2

c) graph the quadratic function f (x) = x2 – 8x + 15; roots for y = 0 are 3 and 5


2 675122

x⎛ ⎞⎜ ⎟⎝ ⎠

− =


BLM 4–4 Section 4.1 Extra Practice 1. a) 2 b) none c) 2 d) 1 2. a) –3, 2 b) no real roots c) – 8.2, 1.2 d) 3 3. a) no solution

b) –2

c) 0, –5

d) 2, –2

4. a) 1.1, –3.5 b) –3.9, 3.9 c) no solution d) –2.8, 1.8 5. Example: a) –10, 15 b) – 20, 20

c) – 0.7, 0.1 d) no solution 6. a) m = 16 b) m < 16 c) m > 16 7. 4.5 s 8. 5 cm, 12 cm, 13 cm

BLM 4–5 Section 4.2 Extra Practice 1. a) (x + 4)(x – 5) b) 3(x – 3)(x – 7) c) –4(x + 1)(x + 2) d) 1

2 ( 3)( 4)x x+ −

2. a) (2x – 1)(7x + 5) b) (x + 5)(3x – 4) c) (4x + 3y)(x + y) d) (2x – 3)(3x – 4) 3. a) 4(3x + 2y)(x – y) b) 3y(2x + 5)(x + 2) c) 10(7x – 5y)(2x – 5y) d) 7x(3x + y)(2x + 3y) 4. a) (x – 7y)(x + 7y) b) (5x – 3)(5x + 3)

c) 52

x y⎛ ⎞+⎜ ⎟⎝ ⎠

52

x y⎛ ⎞−⎜ ⎟⎝ ⎠

or 14 (2 5 )(2 5 )x y x y+ −

d) 16(x – 3) 5. a) (x + 4)(x – 8) b) (6x + 7)(4x – 3) c) 2(7x + 4)(7x – 3) d) (2x2 + 3)(x2 – 3) 6. a) –3, 5 b) 4, –8 c) 3, 6 d) 5±

7. a) 12

− , 43

b) 5, 17

− c) 15

− , 2 d) 32

, –6

8. a) 138

, 138

− b) 73

, 73

− c) 14

, 14

− d) 8, –10

9. a) –1, 23

b) 12

, 4 c) 13

− , 12

d) 6, 72

−

10. a) 13

− b) 32

c) 52

− d) 47

BLM 4–6 Section 4.3 Extra Practice

1. a) 36 b) 100 c) 494

d) 425

2. a) (x + 3)2 = 5 b) (x – 4)2 = 11 c) d) (x + 5)2 = 33 3. a) –1, 9 b) 0, –1 c) 0.9, –0.7 d) –7.5, –6.5

4. a) 31− ± b) 5 132

± c) 0.2, –0.8 d) 37

5. a) 34

, –1 b) 23

1− ± c) 62 2− ± d) 1, 5

6. a) –0.21, 4.71 b) –0.26, 1.26 c) –9.47, –0.53 d) –0.88, 0.38 7. 6, 16

BLM 4–7 Section 4.4 Extra Practice 1. a) two real roots b) no real roots c) one real root d) no real roots 2. a) none b) 1 c) 2 d) 2

3. a) 5 ± 2 b) 7 32± c) 2

3 d) 0, 3

2

4. a) 0.50, 0.33 b) no solution c) –0.59, 2.26 d) –4.46, 1.12



5. a) 52− ± b) 1 2 22

± c) 5 34

− ± d) 72 ±

6. a) No solution;

b) 0, –7; Factor method: can be factored quickly because x is a common factor

c) 5;

2− Factor method: a perfect square trinomial

d) 34 ;− ± Complete the square method: already in the form (x + a)2 = b

e) 1 76

;− ± Quadratic formula: exact values are

required for the answer

7. a) –3 b) 12

−

BLM 4–8 Chapter 4 Review #22

( )

2 2

2 2

2

2

2

2

2

2

2

2

4 4

42

42 4

42

b ca a

b b b ca aa a

b b aca a

b b aca a

b b aca

ax bx c

x x

x x

x

x

x

2

−

4

−

− ± −

+ = −

+ = −

+ + = −

+ =

+ = ±

=

Subtract c from both sides. Divide both sides by a. Complete the square. Factor the perfect square trinomial. Take the square root of both sides. Solve for x.

BLM 4–9 Chapter 4 Test

1. A 2. B 3. D 4. B 5. A

6. 5 52

or 5.59 s

7. a) In line 2, –4 should be in brackets. 2 10.

2±

b) In step 3, each term should have been divided

by 15. 3 3915

− ± .

8. a) x = 2 or 8; Example: Factoring, because the equation is easily factored to (x – 2)(x – 8).

b) x = –7 or 2;

3x = Example: Quadratic formula,

because the equation is not readily factored. c) 23 ;x = ± Example: Completing the square, because it is easy to find the perfect square. d) x = 1 or 5; Example: Determining square roots, because it is easy to find the roots for (x – 3)2 = 4

9. x2 + 5x – 10 = 0; 5 652

− ±

10. 52

k >

11. 11.3 m by 9.3 m

12. 23

or 32

13. 2.57 s

BLM U2–4 Unit 2 Test 1. A 2. B 3. C 4. B 5. B

6. A 7. B 8. 2 9. 13

a = 10. 5

11. a) x = 3.5; (3.5, –7); up; all real numbers; (0, 31.1)

b) y = 289

(x – 3.5)2 – 7

12. (1, –3); all real numbers; y ≥ –3; up; x = 1; x-intercepts have values of 1.87 and 0.13; y-intercept has a value of 1 13. a) 2 m b) 3.67 m c) 8.05 m

chapter 4 prerequisite skills - mrs. lesieur's online...

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