section 1 part 1 chapter 9. copyright © 2012, 2008, 2004 pearson education, inc. 1 objectives 2 5 3...
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Section 1Part 1
Chapter 9
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1
Objectives
2
5
3
4
Part 1 - The Square Root Property
Review the zero-factor property.
Learn the square root property.
Solve quadratic equations of the form (ax + b)2 = c by extending the square root property.
.
9.1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Quadratic Equation
An equation that can be written in the form
where a, b, and c are real numbers, with a ≠ 0, is a quadratic equation. The given form is called standard form.
2 0 ,ax bx c
Slide 9.1- 3
The Square Root Property and Completing the Square
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Review the zero-factor property.
Objective 1
Slide 9.1- 4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Zero-Factor Property
If two numbers have a product of 0, then at least one of the numbers must be 0. That is, if ab = 0, then a = 0 or b = 0.
Slide 9.1- 5
Review the zero-factor property.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the zero-factor property to solve 2x2 − 3x = − 1.
22 3 1 0x x
(2 1)( 1) 0x x
2 1 0 or 1 0x x
2 1x 1x 1
2x
Solution set is 1
,1 .2
Slide 9.1- 6
CLASSROOM EXAMPLE 1
Using the Zero-Factor Property
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Learn the square root property.
Objective 2
Slide 9.1- 7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Square Root Property
If x and k are complex numbers and x2 = k, then
or .x k x k
Remember that if k ≠ 0, using the square root property always produces two square roots, one positive and one negative.
Slide 9.1- 8
Learn the square root property.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve each equation.
2 64m By the square root property, m = 8 or m = –8.
The solution set is {–8, 8}.
23 54 0x 23 54x 2 18x
By the square root property,
18 or 18,x x
3 2 or 3 2.x x
Check:
3 2: 3 18 54 0x
True
Solution set is 3 2, 3 2 .
Slide 9.1- 9
CLASSROOM EXAMPLE 2
Using the Zero-Factor Property
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
An expert marksman can hold a silver dollar at forehead level, drop it, draw his gun, and shoot the coin as it passes waist level. If the coin falls about 4 ft, use the formula d = 16t2 to find the time that elapses between the dropping of the coin and the shot.
d = 16t2
4 = 16t2
By the square root property, Since time cannot be negative, we discard the negative solution. Therefore, 0.5 sec elapses between the dropping of the coin and the shot.
Slide 9.1- 10
CLASSROOM EXAMPLE 3
Using the Square Root Property in an Application
Solution:
1 1 or
2 2t t
21
4t
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve quadratic equations of the form (ax + b)2 = c by extending the square root property.
Objective 3
Slide 9.1- 11
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve (x − 3)2 = 16.
3 16 or 3 16x x
3 4x
7x
3 4 x
1x
Slide 9.1- 12
CLASSROOM EXAMPLE 4
Extending the Square Root Property
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Check:
The solution set is 1,7 .
2( 3) 17 6 24 16
2( 3) 61 1
16 16
True
24 16
Slide 9.1- 13
CLASSROOM EXAMPLE 4
Extending the Square Root Property (cont’d)
True
16 16
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solve (3x + 1)2 = 2.
3 1 2 or 3 1 2x x
3 1 2 x
1 2
3x
3 1 2 x
1 2
3x
Slide 9.1- 14
CLASSROOM EXAMPLE 5
Extending the Square Root Property
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Check:
True
The solution set is 1 2
.3
We show a check for the first solution. The check for the other solution is similar.
(3x + 1)2 = 22
1 2 3 2
31
2
1 2 1 2
2
2 2
2 2
? Simplify.
? Multiply.
? Let x =1 2
.3
Slide 9.1- 15
CLASSROOM EXAMPLE 5
Extending the Square Root Property (cont’d)