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.