# Solving Quadratic Inequalities

How to solve quadratic inequalities: examples and their solutions.

## Example 1: Solve *x*^{2} - 3*x* - 10 ≤ 0

Factor *x*^{2} - 3*x* - 10.

Factor a quadratic trinomial

Find a pair of numbers

whose product is the constant term [-10]

and whose sum is the middle term's coefficient [-3].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

(-1, 10) and (-2, 5)

are not the right numbers.

[-10] = -5⋅2

-5 + 2 = [-3]

So -5 and 2 are the right numbers.

Use -5 and +2

to write a factored form:

(*x* - 5)(*x* + 2) ≤ 0.

Find the zeros of (*x* - 5)(*x* + 2) ≤ 0.

1) *x* - 5 = 0

So *x* = 5.

2) *x* + 2 = 0

So *x* = -2.

So the zeros are *x* = 5, -2.

Solving a quadratic equation by factoring

Roughly draw *y* = (*x* - 5)(*x* + 2)

on the *x*-axis.

Use the zeros 5 and -2.

Quadratic function - Finding zeros

The inequality sign shows that*y* = (*x* - 5)(*x* + 2) ≤ 0.

So color the function below the *x*-axis,

including the zeros: -2, 5.

Then the range of *x* is

-2 ≤ *x* ≤ 5.

## Example 2: Solve *x*^{2} - 4^{2} > 0

16 = 4^{2}

*x*^{2} - 4^{2} = (*x* + 4)(*x* - 4)

Factor the difference of two squares (*a*^{2} - *b*^{2})

Find the zeros of (*x* + 4)(*x* - 4) > 0.

1) *x* + 4 = 0

So *x* = -4.

2) *x* - 4 = 0

So *x* = 4.

So the zeros are *x* = ±4.

Solving a quadratic equation by factoring

Roughly draw *y* = (*x* + 4)(*x* - 4)

on the *x*-axis.

Use the zeros ±4.

The inequality sign shows that*y* = (*x* + 4)(*x* - 4) > 0.

So color the function above the *x*-axis,

excluding the zeros: ±4.

Then *x* < -4 or *x* > 4.

## Example 3: Solve -*x*^{2} + 10*x* - 25 ≥ 0

To make the coefficient of *x*^{2} (+),

multiply both sides by -1.

Don't forget to change

the order of the inequality sign.

Linear inequality (One variable)

-10*x* = -2⋅5⋅*x*

+25 = +5^{2}

*x*^{2} - 2⋅5⋅*x* + 5^{2} = (*x* - 5)^{2}

Factor a perfect square trinomial (*a*^{2} ± 2*ab* + *b*^{2})

Find the zeros of (*x* - 5)^{2} ≤ 0.*x* - 5 = 0

So *x* = 5.

This is the zero.

Roughly draw *y* = (*x* - 5)^{2}

on the *x*-axis.

Use the zero 5.

The inequality sign shows that*y* = (*x* - 5)^{2} ≤ 0.

So color the function below the *x*-axis,

including the zero: 5.

Only *x* = 5 is in the range.

So *x* = 5 is the answer.