Solving Quadratic Equations – The Discriminant

• The Discriminant is the expression found under the radical symbol in the quadratic formula.

• Discriminant =

2 4

2

b b acx

a

2 4b ac

Three possible solutions of a quadratic equation.

Two Different Real Solutions

2 4 0b ac • Case I: (discriminant is positive)

2 10

2x

Example:

2 10 2 10,

2 2x

Discriminant

One Repeated Real Solution

2 4 0b ac • Case II: (discriminant is zero)

2 0

2x

Example:

1, 1x 2 0 2 0,

2 2x

Two Different Imaginary Solutions

2 4 0b ac • Case III: (discriminant is negative)

2 4

2x

Example:

1 , 1x i i 2 2

2

ix

Two Different Imaginary Solutions

2 4 0b ac • Case III:

2 4 0b ac • Case I:

Two Different Real Solutions

2 4 0b ac • Case II:

One Repeated Real Solution

Summary

• Example 1:

Move all terms to left side

Determine the type and number of solutions for the quadratic equation.

2 12x x

2 12 0x x

Determine the values of a, b, and c

1, 1, 12a b c

Determine the value of the discriminant

1, 1, 12a b c

21 4(1)( 12) 2 4b ac

1 48 49 0

Two Different Real Solutions