Solving Quadratic Inequalities

Solving Quadratic Inequalities

How to solve quadratic inequalities problems: examples and their solutions.

Example 1

Solve the given inequality. x^2 - 3x - 10 <= 0

Find the roots of x2 - 3x - 10 = 0:
x = -2, 5.

Solving a quadratic equation by factoring

Draw the x-axis.
Point x = -2, 5.
And roughly draw y = (x + 2)(x - 5).

Quadratic function: factored form

(x + 2)(x - 5) ≤ 0
(left side) ≤ 0

So draw full circles on x = -2 and 5.

And draw the region on the x-axis
where the function is below the x-axis.

Then -2 ≤ x ≤ 5.

Example 2

Solve the given inequality. x^2 - 16 > 0

Find the roots of x2 - 16 = 0:
x = ±4.

Solving a quadratic equation by factoring

Draw the x-axis.
Point x = -4, 4.
And roughly draw y = (x + 4)(x - 4).

Quadratic function: factored form

(x + 4)(x - 4) > 0
(left side) > 0

So draw empty circles on x = -4 and 4.

And draw the region on the x-axis
where the function is above the x-axis.

Then x < -4 or x > 4.

Example 3

Solve the given inequality. -x^2 + 10x - 25 >= 0

The coefficient of x2 is (-).

So multiply -1 on both sides
to change it to (+).

And change the order of the inequality sign:
≥ → ≤

Find the root of x2 - 10x + 25 = 0:
x = 5.

Solving a quadratic equation by factoring

Draw the x-axis.
Point x = 5.
And roughly draw y = (x - 5)2.

Quadratic function: factored form

(x - 5)2 ≤ 0 (Use the changed inequality sign: ≤)
(left side) ≤ 0

So draw a full circle on x = 5.

And draw the region on the x-axis
where the function is below the x-axis.

Only x = 5 satisfies this condition.

So x = 5.