CHAPTER

9Quadratic Equations

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9.1 Introduction to Quadratic Equations

9.2 Solving Quadratic Equations by Completing the Square

9.3 The Quadratic Formula

9.4 Formulas

9.5 Applications and Problem Solving

9.6 Graphs of Quadratic Equations

9.7 Functions

OBJECTIVES

9.2 Solving Quadratic Equations by Completing the Square

Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

a Solve quadratic equations of the type ax2 = p.b Solve quadratic equations of the type (x + c)2 = dc Solve quadratic equations by completing the square.d Solve certain applied problems involving quadratic

equations of the type ax2 = p.

The equation x2 = d has two real solutions when d > 0. The solutions are

The equation x2 = d has no real-number solution when d < 0.

The equation x2 = 0 has 0 as its only solution.

and .d d

9.2 Solving Quadratic Equations by Completing the Square

The Principle of Square Roots

Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolution We use the principle of square roots:

x2 = 25

x = 5 or x = –5

We check mentally that 52 = 25 and (–5)2 = 25. The solutions are 5 and –5.

25 or 25x x

9.2 Solving Quadratic Equations by Completing the Square

a Solve quadratic equations of the type ax2 = p.

A Solve: x2 = 25

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EXAMPLESolutiona)

Check: x2 = 19 x2 = 19 19 19 19 = 19 19 = 19

The solutions are

2 19x 19 or 19x x

2

19 2

19

19 and 19.

9.2 Solving Quadratic Equations by Completing the Square

a Solve quadratic equations of the type ax2 = p.

B Solve: a) x2 = 19 b) 9x2 = 27 c) 5x2 60 = 0

(continued)

Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLESolutionb) c)

29 27x 2 3x

3 or 3x x

25 60 0x 25 60x 2 12x

12 or 12x x

2 3 or 2 3x x

9.2 Solving Quadratic Equations by Completing the Square

a Solve quadratic equations of the type ax2 = p.

A Solve: a) x2 = 19 b) 9x2 = 27 c) 5x2 60 = 0

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EXAMPLESolutiona)

x – 3 = 4 or x – 3 = –4 x = 7 or x = –1

The solutions are 7 and –1. We leave the check to the student.

2( 3) 16x 2( 3) 16 x

9.2 Solving Quadratic Equations by Completing the Square

b Solve quadratic equations of the type (x + c)2 = d

C Solve: a) (x – 3)2 = 16 b) (x + 3)2 = 5

(continued)

Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLEb) (x + 3)2 = 5

The solutions check and can be written as (read as “negative three plus or minus the

square root of five”).

3 5 or 3 5x x

3 5 or 3 5x x

3 5

9.2 Solving Quadratic Equations by Completing the Square

b Solve quadratic equations of the type (x + c)2 = d

C Solve: a) (x – 3)2 = 16 b) (x + 3)2 = 5

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EXAMPLESolution

x2 + 8x + 16 = 17 (x + 4)2 = 17

4 17 or 4 17x x

4 17 or 4 17x x

Sometimes we can factor an equation to express it as a square of a binomial.

9.2 Solving Quadratic Equations by Completing the Square

b Solve quadratic equations of the type (x + c)2 = d

D Solve: x2 + 8x + 16 = 17

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To complete the square for an expression like x2 + bx, we take half of the coefficient of x and square it. Then we add that number, which is (b/2)2.

9.2 Solving Quadratic Equations by Completing the Square

Completing the Square

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EXAMPLESolutionThe coefficient of x is –16.

Half of –16 is –8 and (–8)2 = 64Thus, x2 – 16 becomes a perfect-square trinomial

when 64 is added:x2 – 16x + 64 is the square of x – 8

The number 64 completes the square. Check: (x – 8)2 = (x – 8)(x – 8) = x2 – 8x – 8x + 64 = x2 – 16x + 64

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

E Complete the square: x2 – 16x

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EXAMPLESolutionTake half of 8 and square it to get 16.We add 16 to both sides of the equation.

x2 8x + 16 = 15 + 16 (x - 4) (x - 4) = 1

(x 4)2 = 1x 4 = 1 or x 4 = 1 x = 5 or x = 3

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

F Solve by completing the square. x2 8x = 15

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EXAMPLE

SolutionWe have x2 14x 7 = 0

x2 14x = 7 x2 14x + 49 = 7 + 49

(x 7)2 = 567 56 or 7 56x x

7 2 14 or 7 2 14x x

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

G Solve by completing the square. x2 14x 7 = 0

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EXAMPLE

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

H Solve by completing the square. 3x2 + 7x + 1 = 0

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EXAMPLESolution The coefficient of the x2 term must be 1. When it is not, we must multiply or divide on both sides to find an equivalent equation with an x2 coefficient of 1.

23 7 1 0x x

1 13 3

23 7 1 0 x x

2 7 10

3 3x x 2 7 1

3 3

x x

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

H Solve by completing the square.

(continued)

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EXAMPLE

7 37 7 37 or

6 6 6 6x x

27 37

6 36x

7 37 7 37

or 6 6 6 6

x x

2 49 49

36

7 1

3 3 36 x x

27 12 49

6 36 36

x

27 37

6 36x

9.2 Solving Quadratic Equations by Completing the Square

c Solve quadratic equations by completing the square.

H Solve by completing the square.

Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

To solve a quadratic equation ax2 + bx + c = 0 by completing the square:1. If a ≠ 1, multiply by 1/a so that the x2 –coefficient is 1.2. If the x2 –coefficient is 1, add so that the equation is in the form x2 + bx = –c, or if step (1) has been applied. 3. Take half of the x-coefficient and square it. Add the result on both sides of the equation.4. Express the side with the variables as the square of a binomial.

9.2 Solving Quadratic Equations by Completing the Square

Solving by Completing the Square

(continued)

Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

To solve a quadratic equation ax2 + bx + c = 0 by completing the square:5. Use the principle of square roots and complete the solution.

9.2 Solving Quadratic Equations by Completing the Square

Solving by Completing the Square

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EXAMPLE

1. Familiarize. A formula that fits this situation is s = 16t2, where s is the distance, in feet, traveled by a body falling freely from rest in t seconds. We know that s is 1670 feet.

2. Translate. We know the distance is 1670 feet and that we need to solve for t.

1670 = 16t2

9.2 Solving Quadratic Equations by Completing the Square

d Solve certain applied problems involving quadratic equations of the type ax2 = p.

I The Taipei 101 tower in Taiwan is 1670 feet tall. How long would it take an object to fall to the ground from the top?

(continued)

Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE3. Solve. 1670 = 16t2

21670

16t

1670 1670 or

16 16 t t

10.2 or 10.2 t t

9.2 Solving Quadratic Equations by Completing the Square

d Solve certain applied problems involving quadratic equations of the type ax2 = p.

I

(continued)

Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

EXAMPLE4. Check. The number –10.2 cannot be a solution

because time cannot be negative.s = 16(10.2)2 = 16(104.04) = 1664.64This answer is close.

5. State. It takes about 10.2 seconds for an object to fall to the ground from the top of Taipei 101 .

9.2 Solving Quadratic Equations by Completing the Square

d Solve certain applied problems involving quadratic equations of the type ax2 = p.

I

Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.