The discriminant D = b^{ 2}  4 a c helps solving quadratic inequalities.
Example 1: Solve the inequality
3 x^{ 2} > x + 4
Solution to Example 1:
 Given
3 x^{ 2} > x + 4
 Rewrite the inequality with one side equal
to zero.
3 x^{ 2} + x  4 > 0
 Find the discriminant D.
D = b^{ 2}  4 a c = 1^{ 2}  4 (3) (4) = 49
 Since the discriminant is positive, the left side 3 x^{ 2} + x  4 of the inequality has two zeros at which the sign changes.
 Factor the left side of the inequality.
(3x + 4)(x  1) > 0
 The two real zeros  4 / 3 and 1 of the left side of the inequality, divide the real number line into 3 intervals.
(? ,  4 / 3) ( 4 / 3 , 1) and (1 , +?)
 We chose a real number within each interval and use
it to find the sign of (3x + 4)(x  1).
 a) interval (? ,  4 / 3)
 chose x =  2 and find the sign of (3x + 4)(x  1)
(3x + 4)(x  1) = (3(2) + 4)(2  1)
= 6
(3x + 4)(x  1) is positive in (? ,  4 / 3)
 b) interval ( 4 / 3 , 1)
 chose x = 0 and evaluate (3x + 4)(x  1)
(3x + 4)(x  1) = (0 + 4)(0  1)
=  4
(3x + 4)(x  1) is negative in ( 4 / 3 , 1)
 c) interval (1 , +?)
 chose x = 4 and evaluate (3x + 4)(x  1)
(3x + 4)(x  1) = (3(4) + 4)((4)  1)
= 48
(3x + 4)(x  1) is positive in (1 , +?)
 We need values of x for which (3x + 4)(x  1) is greater than 0, hence the solution set.
( ? ,  4 / 3) U (1 , + ?)
Example 2: Solve the inequality
x^{ 2} < x  4
Solution to Example 2:
 Given
x^{ 2} < x  4
 Rewrite the inequality with one side equal
to zero.
x^{ 2} + x + 4 < 0
 Find the discriminant D.
D = b^{ 2}  4 a c = 1^{ 2}  4 (1) (4) =  15
 Since the discriminant is negative, the left side x^{ 2} + x + 4 of the inequality has no zeros and therefore has the same sign over the interval ( ? , + ?). What we need to do is to find this sign using one test value only
 We chose x = 0 and evalute the left side of the inequality.

x^{ 2} + x + 4 = 0 + 0 + 4
 x^{ 2} + x + 4 is positive in the interval ( ? , + ?) and the given inequality has no solutions.
Exercises: Solve the quadratic inequalities
1. x^{ 2} + 2 x > 3
2. x^{ 2}  4 x > 6
Solutions to Above Exercises:
1. ( 1 , 3)
2. ( ? , + ?)
More references and links on how to Solve Equations, Systems of Equations and Inequalities and Step by Step Solver for Quadratic Inequalities
