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.
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