The Discriminant
In the quadratic formula, the radicand, b2 - 4ac, is called the discriminant
of the quadratic equation ax2 + bx + c = 0.
We can use the discriminant to determine the nature of the solutions of a
quadratic equation without having to solve the equation.
| Discriminant |
Solutions |
|
| b2 - 4ac > 0 |
two different real numbers |
 |
| b2 - 4ac = 0 |
two identical real numbers |
 |
| b2 - 4ac < 0 |
no real number solutions |
 |
If the discriminant is a perfect square, the
solutions will not only be real numbers,
they will also be rational numbers.
Example
Use the discriminant to determine the nature of the solutions of this
quadratic equation: -2x2 - x + 7 = 0
Solution
| The equation has the form ax2 + bx + c = 0 where a
= -2, b = -1, and c = 7. |
|
| Substitute the values of a, b, and c into the discriminant and simplify. |
b2 - 4ac |
= (-1)2 - 4(-2)(7)
= 1 + 56
= 57 |
The discriminant is 57, a positive number.
So the equation -2x2 - x + 7 = 0 has two unequal real number solutions.
|
