# Solving Quadratic Inequalities

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

## Example 1

Find the roots of *x*^{2} - 3*x* - 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

Find the roots of *x*^{2} - 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

The coefficient of *x*^{2} is (-).

So multiply -1 on both sides

to change it to (+).

And change the order of the inequality sign:

≥ → ≤

Find the root of *x*^{2} - 10*x* + 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.