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

9.1 Introduction to Quadratic Equations

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

a Write a quadratic equation in standard formax2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c.

b Solve quadratic equations of the type ax2 + bx = 0, where b ≠ 0, by factoring.

c Solve quadratic equations of the type ax2 + bx + c = 0, where b ≠ 0 and c ≠ 0, by factoring.

d Solve applied problems involving quadratic equations.

The following are quadratic equations. They contain polynomials of second degree.

4x2 + 7x – 5 = 0 3y2 – y = 95a2 = 8a 12m2 = 144

9.1 Introduction to Quadratic Equations

a Write a quadratic equation in standard formax2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c.

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

A quadratic equation is an equation equivalent to an equation of the type

ax2 + bx + c = 0, a > 0,where a, b, and c are real-number constants. We say that the preceding is the standard form on a quadratic equation.

9.1 Introduction to Quadratic Equations

Quadratic Equation

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EXAMPLE

a. 5x2 + 8x – 3 = 0The equation is in standard form.5x2 + 8x – 3 = 0a = 5; b = 8; c = –3

9.1 Introduction to Quadratic Equations

a Write a quadratic equation in standard formax2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c.

A Write in standard form and determine a, b, and c.

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EXAMPLE

b. 6y2 = 5y6y2 – 5y = 0a = 6; b = –5; c = 0

9.1 Introduction to Quadratic Equations

a Write a quadratic equation in standard formax2 + bx + c = 0, a > 0, and determine the coefficients a, b, and c.

A Write in standard form and determine a, b, and c.

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EXAMPLESolution 8x2 + 3x = 0x(8x + 3) = 0 Factoringx = 0 or 8x + 3 = 0 Using the principle of zero productsx = 0 or 8x = –3

x = 0 or 3

8x

9.1 Introduction to Quadratic Equations

b Solve quadratic equations of the type ax2 + bx = 0, where b ≠ 0, by factoring.

B Solve: 8x2 + 3x = 0.

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EXAMPLECheck: 8x2 + 3x = 0

8(0)2 + 3(0) = 00 = 0 True

23 38 88( ) 3( ) 0

9 964 88( ) ( ) 0 9 98 8( ) ( ) 0

Both solutions check.

9.1 Introduction to Quadratic Equations

b Solve quadratic equations of the type ax2 + bx = 0, where b ≠ 0, by factoring.

B Solve: 8x2 + 3x = 0.

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0 = 0True

8x2 + 3x = 0

A quadratic equation of the type ax2 + bx = 0, where c = 0 and b ≠ 0, will always have 0 as one solution and a nonzero number as the other solution.

9.1 Introduction to Quadratic Equations

b Solve quadratic equations of the type ax2 + bx = 0, where b ≠ 0, by factoring.

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EXAMPLESolution : Write the equation in standard form and then try factoring.

(y – 7)(y – 2) = 4y – 22 y2 – 9y + 14 = 4y – 22 Multiplying y2 – 13y + 36 = 0 Standard form

(y – 4)(y – 9) = 0 y – 4 = 0 or y – 9 = 0 y = 4 or y = 9 The solutions are 4 and 9.

9.1 Introduction to Quadratic Equations

c Solve quadratic equations of the type ax2 + bx + c = 0, where b ≠ 0 and c ≠ 0, by factoring.

C Solve: (y – 7)(y – 2) = 4y – 22

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EXAMPLEThe number of diagonals d in a polygon that has n sides is given by the formula

If a polygon has 54 diagonals, how many sides does it have?

2 3.

2

n nd

9.1 Introduction to Quadratic Equations

d Solve applied problems involving quadratic equations.

D Applications of Quadratic Equations

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EXAMPLE1. Familiarize. A sketch can help us to become familiar

with the problem. We draw a hexagon (6 sides) and count the diagonals. As the formula predicts, for n = 6, there are 9 diagonals:

26 63 36 189

2 2

d

9.1 Introduction to Quadratic Equations

d Solve applied problems involving quadratic equations.

D Applications of Quadratic Equations

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EXAMPLE2. Translate. Since the number of diagonals is 54, we

substitute 54 for d:2 3

4 .2

5

n n

9.1 Introduction to Quadratic Equations

d Solve applied problems involving quadratic equations.

D Applications of Quadratic Equations

(continued)

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EXAMPLE3. Solve. We solve the equation for n, first

reversing the equation for convenience.2 3

542

n n

2 3 108n n 2 3 108 0n n

( 12)( 9) 0n n

12 0 or 9 0n n

12 or 9n n

9.1 Introduction to Quadratic Equations

d Solve applied problems involving quadratic equations.

D Applications of Quadratic Equations

(continued)

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EXAMPLE

4. Check. Since the number of sides cannot be negative, –9 cannot be a solution.

5. State. The polygon has 12 sides.

9.1 Introduction to Quadratic Equations

d Solve applied problems involving quadratic equations.

D Applications of Quadratic Equations

Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.