# Quadratic Inequality

How to solve a quadratic inequality: 3 examples and their solutions.

## Examplex^{2} - 3x - 10 ≤ 0

First, find the zeros of the left side.

Quadratic Function: Find Zeros

Factor the left side

x^{2} - 3x - 10.

Find a pair of numbers

whose product is the constant term -10

and whose sum is the coefficient of the middle term -3.

-5⋅2 = -10

-5 + 2 = -3

Then (x - 5)(x + 2) ≤ 0.

Factor a Quadratic Trinomial

Case 2) x + 2 = 0

Then x = -2.

Case 1) x = 5

Case 2) x = -2

So the zeros are x = -2, 5.

Draw y = (x - 5)(x + 2)

on the x-axis.

First point the zeros x = -2 and 5.

And draw a parabola

that passes through x = -2 and 5.

Quadratic Function: Find Zeros

See (x - 5)(x + 2) ≤ 0.

The left side is less than or equal to 0.

So color the region

where the graph is below the x-axis (y = 0).

The inequality sign includes equal to [=].

So draw full circles on the zeros:

x = -2 and x = 5.

The colored region is

-2 ≤ x ≤ 5.

So

-2 ≤ x ≤ 5

is the answer.

## Examplex^{2} - 16 > 0

First, find the zeros of the left side.

-16 = -4^{2}

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

Factor the Difference of Two Squares: a^{2} - b^{2}

Find the zeros.

Case 1) x + 4 = 0

Then x = -4.

Case 2) x - 4 = 0

Then x = 4.

Case 1) x = -4

Case 2) x = 4

So the zeros are x = ±4.

Draw y = (x + 4)(x - 4)

on the x-axis.

First point the zeros x = -4 and 4.

And draw a parabola

that passes through x = -4 and 4.

See (x + 4)(x - 4) ≤ 0.

The left side is greater than 0.

So color the regions

where the graph is above the x-axis (y = 0).

The inequality sign does not include equal to [=].

So draw empty circles on the zeros:

x = -4 and x = 4.

The colored regions are

x < -4 or x > 4.

So

x < -4 or x > 4

is the answer.

## Example-x^{2} + 10x - 25 ≥ 0

Divide both sides by (-).

Dividing both sides by (-)

does change the order of the inequality sign:

≥ → ≤.

Linear Inequality (One Variable)

Make a perfect square trinomial.

x^{2} is x^{2}.

-10x is

-2 times

x times,

(-10x)/(-2⋅x), 5.

+25 is +5^{2}.

x^{2} - 2⋅x⋅5 + 5^{2} = (x + 5)^{2}

Find the zeros.

The zero is x = 5.

Draw y = (x + 5)^{2}

on the x-axis.

First point the zero x = 5.

And draw a parabola

that touches x = 5.

See (x + 5)^{2} ≤ 0.

The left side is less than or equal to 0.

So color the region

where the graph is below the x-axis (y = 0).

But there's no region to color.

Then see the inequality sign.

The inequality sign includes equal to [=].

So draw a full circle on the zero:

x = 5.

The colored region is

x = 5.

So

x = 5

is the answer.