# 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