Section 1 Chapter 11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 3 4 Solving Quadratic Equations by the Square Root Property

Section 1Chapter 11

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

1

Objectives

2

3

4

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.

Solve quadratic equations with solutions that are not real numbers.

11.1


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 11.1- 3

Solving Quadratic Equations by the Square Root Property



Objective 1

Slide 11.1- 4


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 11.1- 5



Solve each equation by the zero-factor property.

2x2 − 3x + 1 = 0 x2 = 25

Solution:

1x 1

2x or

1,1

2

Slide 11.1-6

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

CLASSROOM EXAMPLE 1


Objective 2


Slide 11.1-7



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 11.1-8

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 15x

22 90 0x 23 8 88x

2 49z 7z

22 90x

7

2 15x 15x 15

23 96

3 3

x

2 32x 4 2x 4 2

Slide 11.1-9

Solving Quadratic Equations of the Form x2 = kCLASSROOM EXAMPLE 2

2 45x 45x

3 53 5x


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


Objective 3


Slide 11.1-11


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 11.1-12


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 11.1-13

Solving Quadratic Equations of the Form (x + b)2 = kCLASSROOM EXAMPLE 4


Solve (5m + 1)2 = 7.

Solution:

1 7

5

5 1 7m 5 1 7m or

5 1 7m 5 1 7m or

1 7

5m

1 7

5m

or

Slide 11.1-14

Solving a Quadratic Equation of the Form (ax + b)2 = kCLASSROOM EXAMPLE 5


Solve quadratic equations with solutions that are not real numbers.

Objective 4

Slide 11.1- 15


Solve the equation.

2 17x

17 or 17 x x

17 or 17x i x i

The solution set is 17 . i

Slide 11.1- 16

CLASSROOM EXAMPLE 6

Solve for Nonreal Complex Solutions

Solution:


25 100x

5 100 or 5 100 x x

5 10 or 5 10 x i x i

5 10 or 5 10 x i x i

The solution set is 5 10 . i

Slide 11.1- 17

CLASSROOM EXAMPLE 6

Solve for Nonreal Complex Solutions (cont’d)

Solve the equation.

Solution:

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Section 1 Chapter 11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives 2 3 4 Solving Quadratic Equations by the Square Root Property