Solve x^2 - 2x - 8 <= 0 Step by Step | Quadratic inequalities

Step by step

The question

Question: solve x² − 2x − 8 ≤ 0.

There is an x², so the answer can be two parts.

x² − 2x − 8 ≤ 0

1 Find two numbers

We need two numbers.

They multiply to −8, and they add to −2.

? × ? = −8

? + ? = −2

They multiply to a minus number, so one number is plus and one is minus.

They add to a minus number, so the bigger one is minus.

× is − → one +, one −; bigger one −

We list the pairs that make 8, smallest first.

We take the smaller from the bigger.

We need 2.

2 and 4 give 2.

Yes!

With the signs: 2 and −4.

1 × 8: 8 − 1 = 7

2 × 4: 4 − 2 = 2 ✓

The two numbers go in the brackets.

So we get (x + 2)(x − 4) ≤ 0.

(x + 2)(x − 4) ≤ 0

2 Find the critical points

A critical point is an x where the side with x is 0, or has no value.

Only there can its sign change.

critical point: where it is 0 (or ÷ 0)

We make each bracket 0.

x + 2 = 0 x − 4 = 0

In x + 2 = 0, we take away 2 on both sides.

In x − 4 = 0, we add 4 on both sides.

x + 2 − 2 = 0 − 2 x − 4 + 4 = 0 + 4

The critical points are −2 and 4.

x = −2 x = 4

3 Make a sign table

Between the critical points the sign stays the same.

So one test number tells us the sign of each part.

We draw a table.

The critical points −2 and 4 split the number line into 3 parts.

At −2, x + 2 is 0.

So the product is 0 there.

At 4, x − 4 is 0.

So the product is 0 there.

Left of −2, we test x = −3.

−3

x + 2 is −1 and x − 4 is −7.

−1

−7

Minus times minus is plus.

Between −2 and 4, we test x = 0.

x + 2 is 2 and x − 4 is −4.

−4

Plus times minus is minus.

Right of 4, we test x = 5.

x + 2 is 7 and x − 4 is 1.

Plus times plus is plus.

4 Read the answer

Less than or equal to 0 means negative or 0.

So we want the minus parts, and the 0s.

So −2 ≤ x ≤ 4.

−2 ≤ x ≤ 4

≤ 0 → the − parts and the 0s

Careful!

Not the plus parts!

Less than 0 means minus.

Careful: not x ≤ −2 or x ≥ 4 ✗

5 Show it on a number line

Books also write the answer like this.

It is called interval notation.

A square bracket means the end is in, like the filled circle.

interval: [−2, 4]

6 Check

Let's check with a picture: the graph of y = x² − 2x − 8.

It is a parabola.

It crosses the x-axis at the critical points, −2 and 4.

Below the x-axis, y is minus.

That is −2 ≤ x ≤ 4: our answer.

Let's check: a number from the answer, 0, put into the question.

x = 0: 0² − 2(0) − 8 = 0 + 0 − 8 = −8

−8 is less than or equal to 0.

True!

−8 ≤ 0 true

Let's look back.

First: find two numbers.

Then: find the critical points.

Then: make a sign table.

Then: read the answer.

Then: show it on a number line.

Last: check.

Answer