Discriminants and determining the number of real roots of a quadratic equation

What is a discriminant?

 

A discriminant is a value calculated from a quadratic equation. It use it to 'discriminate' between the roots (or solutions) of a quadratic equation. 

 

A quadratic equation is one of the form: ax² + bx + c

 

The discriminant, D = b² - 4ac

 

Note: This is the expression inside the square root of the quadratic formula

 

 

There are three cases for the discriminant;

 

Case 1:

 

b² - 4ac > 0

 

If the discriminant is greater than zero, this means that the quadratic equation has two real, distinct (different) roots.

 

Example

 

x² - 5x + 2 = 0

a = 1, b = -5, c = 2

 

Discriminant, D = b² - 4ac

                         = (-5)² - 4 * (1) * (2)

                         = 17

 

Therefore, there are two real, distinct roots to the quadratic equation 

x² - 5x + 2.

 

Case 2:

 

b² - 4ac < 0

 

If the discriminant is greater than zero, this means that the quadratic equation has no real roots.

 

Example

 

3x² + 2x + 1 = 0

a = 3, b = 2, c = 1

 

Discriminant, D = b² - 4ac

                         = (2)^{2 -} 4 * (3) * (1)

                         = - 8

Therefore, there are no real roots to the quadratic equation 3x² + 2x + 1.

 

Case 3:

 

b² - 4ac = 0

 

If the discriminant is equal to zero, this means that the quadratic equation has two real, identical roots.

 

Example

 

x² + 2x + 1 = 0

a = 1, b = 2, c = 1

 

Discriminant, D = b² - 4ac

                         = (2)²  - 4 * (1) * (1)

                         = 0

Therefore, there are two real, identical roots to the quadratic equation x² + 2x + 1.

 

Summary

 

Quadratic equation is ax² + bx + c

Determinant D = b² - 4ac

 

D > 0 means two real, distinct roots.

D = 0 means two real, identical roots/

D < 0 means no real roots.

 

Now try these, (take care with minus signs)

 

Questions

 

Q1. x² - 7x + 2 = 0

Q2. - 3x² + 2x - 1 = 0

Q3. 9x² - 12x + 4 = 0

Q4. - x² + x + 1 = 0

 

Answers

 

Q1. D = 41, means two real, distinct roots.

Q2. D = -16, means no real roots.

Q3. D = 0, means two real, identical roots.

Q4. D = 5, means two real, distinct roots.