Quadratic Inequalities

Tutorial on solving quadratic inequalities with examples and detailed solutions.

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