Factor x^3 - x^2 - 12x = 0

Step by step

The question

Question: solve x³ − x² − 12x = 0.

The right side is 0 already.

1 Factor completely

Common factor first.

Every term has x.

So we take out x.

So it is x(x² − x − 12) = 0.

Careful!

Don't divide both sides by x.

Then the answer x = 0 would be lost.

Now the bracket.

We need two numbers.

They multiply to −12, and they add to −1.

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.

We list the pairs that make 12, smallest first.

We take the smaller from the bigger.

We need 1.

3 and 4 give 1.

Yes!

With the signs: 3 and −4.

So it is x(x + 3)(x − 4) = 0.

2 Each factor is 0

Some numbers multiply to 0.

That only happens if one of them is 0.

So we make each factor 0.

Each factor with x gives a small equation.

x = 0 is done already.

In x + 3 = 0, we take away 3 on both sides.

In x − 4 = 0, we add 4 on both sides.

So x = 0, x = −3 and x = 4.

Three answers.

3 Check

Let's check: put each answer into x³ − x² − 12x = 0.

First x = 0.

Now x = −3.

Now x = 4.

We get 0 every time.

So all three answers are right!

Let's look back.

First: factor completely.

Then: each factor is 0.

Last: check.