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.
