Chapter 9 Section 1. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving...

Chapter 9 Section 1

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solving Quadratic Equations by the Square Root Property

Review the zero-factor property.

Solve equations of the form x2 = k, where k > 0.

Solve equations of the form (ax + b)2 = k, where k > 0.

Use formulas involving squared variables.

9.1

2

3

4


Solving Quadratic Equations by the Square Root PropertyIn Section 6.5, we solved quadratic equations by factoring. Since not all quadratic equations can easily be solved by factoring, we must develop other methods.

Slide 9.1-3


Objective 1

Review the zero-factor property.

Slide 9.1-4


Solving Quadratic Equations by the Square Root PropertyRecall that a quadratic equation is an equation that can be written in the form

for real numbers a, b, and c, with a ≠ 0. We can solve the quadratic equation x2 + 4x + 3 = 0 by factoring, using the zero-factor property.

2 0ax bx c

Slide 9.1-5

Zero-Factor PropertyIf a and b are real number and if ab = 0, then a = 0 or b = 0.

Standard Form


Solve each equation by the zero-factor property.

2x2 − 3x + 1 = 0 x2 = 25

Solution:

1x 12

x or

1 ,12

Slide 9.1-6

EXAMPLE 1 Solving Quadratic Equations by the Zero-Factor Property

5,5

5 or 5x x

5 0 or 5 0x x

5 5 0x x

2 25 0x 2 1 1 0x x 2 1 1 0x x

1 0x 2 1 0x or1 0x 2 1 0x or


Objective 2


Slide 9.1-7



We might also have solved x2 = 9 by noticing that x must be a number whose square is 9. Thus,

or

This can be generalized as the square root property.

9 3.x 9 3x

Slide 9.1-8

Square Root PropertyIf k is a positive number and if x2 = k, then

or

The solution set is which can be written (± is read “positive or negative” or “plus or minus.”)

x k x k

, ,k k .k

When we solve an equation, we must find all values of the variable that satisfy the equation. Therefore, we want both the positive and negative square roots of k.


Solve each equation. Write radicals in simplified form.

Solution:

2 49z 2 12x

2 169x 23 8 88x

2 49z 7z

2 169x

7

2 12x 2 3x 2 3

23 963 3x

2 32x

4 2x 4 2Slide 9.1-9

EXAMPLE 2 Solving Quadratic Equations of the Form x2 = k


Objective 3


Slide 9.1-10


In each equation in Example 2, the exponent 2 appeared with a single variable as its base. We can extend the square root property to solve equations in which the base is a binomial.


Slide 9.1-11


Solve (p – 4)2 = 3.

Solution:

4 3

4 3p 4 3p or

4 44 3p 344 4p or

4 3p 4 3p or

Slide 9.1-12

EXAMPLE 3 Solving Quadratic Equations of the Form (x + b)2 = k


Solve (5m + 1)2 = 7.

Solution:

1 75

5 1 7m 5 1 7m or

5 1 7m 5 1 7m or

1 75

m

1 75

m or

Slide 9.1-13

EXAMPLE 4 Solving a Quadratic Equation of the Form (ax + b)2 = k


Solve (7z + 1)2 = –1.

Solution:

7 1 1z 7 1 1z or

Slide 9.1-14

EXAMPLE 5 Recognizing a Quadratic Equation with No Real Solutions


Objective 4

Use formulas involving squared variables.

Slide 9.1-15


Use the formula, to approximate the length of a bass

weighing 2.80 lb and having girth 11 in.

Solution :

2

,1200L gw

2 1

121

2.801200

00 1200L

2311 11360 11L

2305.5 L

17.48L

The length of the bass is approximately 17.48 in.

Slide 9.1-16

EXAMPLE 6 Finding the Length of a Bass

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Chapter 9 Section 1. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving...

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