Warm-up• 1. Solve the following quadratic equation by

Completing the Square:

• x2 - 10x + 15 = 0

• 2. Convert the following quadratic equation to vertex format

• y = 2x2 – 8x + 20

5 10x

22( 2) 12y x

Chapter 4

Section 4-8

The Discriminant

Objectives

• I can calculate the value of the discriminant to determine the number and types of solutions to a quadratic equation.

Quadratic Review

• Quadratic Equation in standard format:

• y = ax2 + bx + c

• Solutions (roots) are where the graph crosses or touches the x-axis.

• Solutions can be real or imaginary

Types of Solutions

Complex Number System

Real Numbers Imaginary Numbers

Rational Irrationala bi

Types of Solutions

2 Real Solutions

1 Real Solution

2 Imaginary Solutions

2 4

2

b b acx

a

2What value of b -4ac gives each

solution type?

Key Concept for this Section

• What happens when you square any number like below:

• x2 = ?

• It is always POSITIVE!!

• This is always the biggest mistake in this section

Key Concept #2

• What happens when you subtract a negative number like below:

• 3 - -4 = ?

• It becomes ADDITION!!

• This is 2nd biggest error on this unit!

The Quadratic Formula

• The solutions of any quadratic equation in the format ax2 + bx + c = 0, where a 0, are given by the following formula:

• x = a

acbb

2

42

The quadratic equation must be set equal to ZERO before using this formula!!

Discriminant

• The discriminant is just a part of the quadratic formula listed below:

b2 – 4ac• The value of the discriminant determines the

number and type of solutions.

Discriminant PossibilitiesValue of

b2-4acDiscriminant is a Perfect

Square?

# of Solutions

Type of Solutions

> 0 Yes 2 Rational

> 0 No 2 Irrational

< 0 2 Imaginary

= 0 1 Rational

Example 1

• What are the nature of roots for the equation:

• x2 – 8x + 16 = 0

• a = 1, b = -8, c = 16

• Discriminant: b2 – 4ac

• (-8)2 – 4(1)(16)

• 64 – 64 = 0

• 1 Rational Solution

Example 2

• What are the nature of roots for the equation:

• x2 – 5x - 50 = 0

• a = 1, b = -5, c = -50

• Discriminant: b2 – 4ac

• (-5)2 – 4(1)(-50)

• 25 – (-200) = 225, which is a perfect square

• 2 Rational Solutions

Example 3

• What are the nature of roots for the equation:

• 2x2 – 9x + 8 = 0

• a = 2, b = -9, c = 8

• Discriminant: b2 – 4ac

• (-9)2 – 4(2)(8)

• 81 – 64 = 17, which is not a perfect square

• 2 Irrational Solutions

Example 4

• What are the nature of roots for the equation:

• 5x2 + 42= 0

• a = 5, b = 0, c = 42

• Discriminant: b2 – 4ac

• (0)2 – 4(5)(42)

• 0 – 840 = -840

• 2 Imaginary Imaginary

GUIDED PRACTICE for Example 4

Find the discriminant of the quadratic equation and give the number and type of solutions of the equation.

4. 2x2 + 4x – 4 = 0

SOLUTION

Equation Discriminant Solution(s)

ax2 + bx + c = 0 b2 – 4ac

2x2 + 4x – 4 = 0 42 – 4(2)(– 4 )

x =– b+ b2– 4ac2ac

= 48Two irrational solutions

GUIDED PRACTICE for Example 4

5.

SOLUTION

Equation Discriminant Solution(s)

ax2 + bx + c = 0 b2 – 4ac

122 – 4(12)(3 )

x =– b+ b2– 4ac2ac

= 0

One rational solution

3x2 + 12x + 12 = 0

3x2 + 12x + 12 = 0

6.

SOLUTION

Equation Discriminant Solution(s)

ax2 + bx + c = 0 b2 – 4ac x =– b+ b2– 4ac2ac

GUIDED PRACTICE for Example 4

8x2 = 9x – 11

8x2 – 9x + 11 = 0 (– 9)2 – 4(8)(11 )

= – 271

Two imaginary solutions

7.

SOLUTION

Equation Discriminant Solution(s)

ax2 + bx + c = 0 b2 – 4ac x =– b+ b2– 4ac2ac

GUIDED PRACTICE for Example 4

7x2 – 2x = 5

(– 2)2 – 4(7)(– 5 )

= 144

Two rational solutions

7x2 – 2x – 5 = 0

Homework

• WS 7-2