Exploring Quadratic Functions and Inequalities

Advanced Algebra

Chapter 6

Solving Quadratic Functions

Solve the following equation.

254 2 x

54 x

254 2 xSolution:

9x 1x

54 x 54 x

Square of a Binomial

Solving Quadratic Functions

Multiply the following expressions.

Is there a pattern?

Shortcut Method

( x + 6 )2 = x2 + 12x + 36

33 xx

5252 xx

962 xx

25204 2 xx

1st term last term square of1st term

square oflast term

2×product of both terms

Solving Quadratic Functions

Try using the shortcut method with these.

Now Try Backwards:

x2 + 8x + 16 = ( )2

x2 – 4x + 4 = ( )2

x2 + x + ¼ = ( )2

122 xx9

4

3

42 xx 21x

2

3

2x

4x

2x

x + ½

Solving Quadratic Functions by Completing the Square

For example, solve the following equation by

completing the square.

Step 1 Move the constant to the other side.

Step 2 Notice the coefficient of the linear term is 3,

or b = 3. Therefore, is the new constant needed to

create a Square Binomial. Add this value to both sides.

18___32 xx

2

2

b

01832 xx

222

2

318

2

33

xx

Solving Quadratic Functions by Completing the Square

Step 3 Factor and Solve.22

2

2

318

2

33

xx

4

9

4

72

2

32

x

2

9

2

3

x

4

81

2

32

x

2

9

2

3x

2

9

2

3x

2

3

2

9x

2

3

2

9x

32

6x 6

2

12x

Quadratic Formula

Another way to solve quadratic equations is to

use the quadratic formula.

This is derived from the standard form of the

equation ax2 + bx + c = 0 by the process of

completing the square.

a

acbbx

2

42

Quadratic Formula

The Quadratic Formula

The value of the discriminant, b2 – 4ac,

determines the nature of the roots of a quadratic

equation.

a

acbbx

2

42

The Discriminant

Discriminant b2 – 4ac

Value

Description

Sample Graph

-5 5-1

10

x

y

-5 5-1

10

x

y

-5 5

-6

5

x

y

-5 5

-5

5

x

y

b2 – 4ac = 0 b2 – 4ac < 0 b2 – 4ac > 0b2 – 4ac

is a perfectsquare

Intersects the x-axis once. One real root.

Does not intersect the x-axis. Two imaginary roots.

Intersects the x-axis twice. Two real, irrational roots.

Intersects the x-axis twice. Two real, rational roots.

Solving Quadratic Functions with the Quadratic Formula

For example, solve the following equation with

the quadratic formula.

Step 1 Write quadratic equation in Standard Form.

Step 2 Substitute coefficients into quadratic formula.

In this case a = 4, b = –20 and c = 25

025204 2 xx

xx 20254 2

42

25442020 2 x

2

5

8

20x

The discriminant, (–20)2 – 4(4)(25) = 0.There is one real, rational root.

Solving Quadratic Functions with the Quadratic Formula

For example, solve the following equation with

the quadratic formula.

Step 1 Write quadratic equation in Standard Form.

Step 2 Substitute coefficients into quadratic formula.

In this case a = 3, b = –5 and c = 2

0253 2 xx

xx 523 2

32

23455 2 x 1

6

6x

The discriminant, (–5)2 – 4(3)(2) = 1.There are two real, rational roots.

3

2

6

4x

Homework