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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.
(-infinity , - 4 / 3) (- 4 / 3 , 1) and (1 , +infinity)
- We chose a real number within each interval and use
it to find the sign of (3x + 4)(x - 1).
- a) interval (-infinity , - 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 (-infinity , - 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 , +infinity)
- 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 , +infinity)
- We need values of x for which (3x + 4)(x - 1) is greater than 0, hence the solution set.
(- infinity , - 4 / 3) U (1 , + infinity)
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 (- infinity , + infinity). 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 (- infinity , + infinity) 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. (- infinity , + infinity)
More references and links on how to Solve Equations, Systems of Equations and Inequalities.
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