Factor x^3 + 5x^2 + 6x = 0

Step by step

The question

Question: solve x³ + 5x² + 6x = 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 + 5x + 6) = 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 6, and they add to 5.

They multiply to a plus number, so the two numbers have the same sign.

They add to a plus number, so both are plus.

We list the pairs that make 6, smallest first.

We add each pair.

We need 5.

2 and 3 give 5.

Yes!

So it is x(x + 2)(x + 3) = 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 + 2 = 0, we take away 2 on both sides.

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

So x = 0, x = −2 and x = −3.

Three answers.

3 Check

Let's check: put each answer into x³ + 5x² + 6x = 0.

First x = 0.

Now x = −2.

Now x = −3.

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.