Quadratic inequalities

A quadratic inequality is an inequality where the unknown appears in the second power.

It looks like a quadratic equation, but instead of an equals sign we use an inequality sign.

Example

We look at the following inequality:

$$ \large x^2 - 4 < 0 $$

Here we ask: For which values of $ \large x $ is $ \large x^2 - 4 $ less than 0?

Method

To solve a quadratic inequality, we first find the roots of the corresponding quadratic equation:

$$ \large x^2 - 4 = 0 $$

This gives:

$$ \large x = -2 \quad \text{or} \quad x = 2 $$

The two roots divide the number line into three intervals:

Interval 1: $ \large x < -2 $
Interval 2: $ \large -2 < x < 2 $
Interval 3: $ \large x > 2 $

Determine where the inequality holds

We choose a test value in each interval and check whether the inequality is true:

For $ \large x = -3 $:

$ \large (-3)^2 - 4 = 9 - 4 = 5 \quad $ Not less than 0.

For $ \large x = 0 $:

$ \large 0^2 - 4 = -4 \quad $ Less than 0 = the interval applies.

For $ \large x = 3 $:

$ \large 3^2 - 4 = 9 - 4 = 5 \quad $ Not less than 0.

The solution is therefore:

$$ \large -2 < x < 2 $$

Example with discriminant

We look at the inequality:

$$ \large x^2 + x - 6 < 0 $$

First we find the discriminant:

$$ \large D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-6) = 1 + 24 = 25 $$

The roots of the corresponding quadratic equation are:

$$ \large x = \frac{-1 \pm \sqrt{25}}{2} = \frac{-1 \pm 5}{2} $$

$$ \large x = -3 \quad \text{or} \quad x = 2 $$

We now examine the three intervals:

For $ \large x = -4 $:

$ \large (-4)^2 + (-4) - 6 = 16 - 4 - 6 = 6 \quad $ Not less than 0.

For $ \large x = 0 $:

$ \large 0^2 + 0 - 6 = -6 \quad $ Less than 0 = the interval applies.

For $ \large x = 3 $:

$ \large 3^2 + 3 - 6 = 9 + 3 - 6 = 6 \quad $ Not less than 0.

The solution is therefore:

$$ \large -3 < x < 2 $$

Example with separate intervals

We now look at the inequality:

$$ \large x^2 - 4 > 0 $$

The roots are the same: $ \large x = -2 $ and $ \large x = 2 $. The number line is divided into the same three intervals.

We test again:

For $ \large x = -3 $:

$ \large (-3)^2 - 4 = 5 \quad $ Greater than 0 = the interval applies.

For $ \large x = 0 $:

$ \large 0^2 - 4 = -4 \quad $ Not greater than 0.

For $ \large x = 3 $:

$ \large 3^2 - 4 = 5 \quad $ Greater than 0 = the interval applies.

The solution is therefore two separate intervals:

$$ \large x < -2 \quad \text{or} \quad x > 2 $$

Example without solution

Some quadratic inequalities have no solution at all. We look at:

$$ \large x^2 + 1 < 0 $$

The function $ \large x^2 + 1 $ is always at least 1, since $ \large x^2 \geq 0 $. It can therefore never be less than 0.

Conclusion: There are no values of $ \large x $ that satisfy the inequality.

General method

Write the inequality in the form $ \large ax^2 + bx + c \; \lessgtr \; 0 $.
Find the discriminant: $ \large D = b^2 - 4ac $.
Calculate the roots of the corresponding quadratic equation.
Examine the intervals that the roots divide the number line into, to determine where the inequality is true.
If the discriminant is negative, there may be no solutions (as in the example without solution).

Summary

Quadratic inequalities resemble quadratic equations, but the solution is one interval or several intervals.
The roots divide the number line into intervals, which are tested with sample values.
The solution can be one interval, two intervals, or none at all.
A negative discriminant can show that no solutions exist.