# The Discriminant
**Example 1**
Use the discriminant to determine the nature of the solutions of this
quadratic equation: 5x^{2} - 3x + 8 = 0
**Solution **
The equation has the form ax^{2} + bx + c = 0 where a = 5, b= -3, and
c = 8.
The equation has the form ax^{2} + bx
+ c = 0 where a = 5, b = -3, and
c = 8. |
Substitute the values of a, b, and c into the discriminant and simplify. |
b^{2} - 4ac |
= (-3)^{2} - 4(5)(8)
= 9 - 160
= -151 |
The discriminant is -151, a negative number.
So the equation 5x^{2} - 3x + 8 = 0 has no real number solutions.
**Example 2**
Use the discriminant to determine the nature of the solutions of this
quadratic equation: 9x^{2} - 6x = -1
**Solution **
To put the equation in standard form,
add 1 to both sides of the equation.
Now the equation has the form ax^{2} + bx + c = 0 where a = 9, b
= - 6, and c = 1. |
9x^{2} - 6x = -1
9x^{2} - 6x + 1 = 0 |
Substitute the values of a, b, and c into
the discriminant and simplify. |
b^{2} - 4ac |
= (-6)2 - 4(9)(1) = 36 - 36
= 0 |
The discriminant is 0.
So the equation 9x^{2} - 6x = -1 has two identical real number solutions. |