Factor 2x^3 + 4x^2 = 30x

Step by step

The question

Question: solve 2x³ + 4x² = 30x.

1 Get 0 on one side

To factor, we want 0 on the right side.

So we take away 30x on both sides.

We do it on both sides, so they stay equal.

Now we have 2x³ + 4x² − 30x = 0.

2 Factor completely

Common factor first.

Every term has x, and every number divides by 2.

So we take out 2x.

So it is 2x(x² + 2x − 15) = 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 −15, and they add to 2.

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

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

We list the pairs that make 15, smallest first.

We take the smaller from the bigger.

We need 2.

3 and 5 give 2.

Yes!

With the signs: −3 and 5.

So it is 2x(x − 3)(x + 5) = 0.

3 Each factor is 0

Some numbers multiply to 0.

That only happens if one of them is 0.

So we make each factor 0.

2 is a plain number.

It is never 0, so it does not make the product 0.

Each factor with x gives a small equation.

x = 0 is done already.

In x − 3 = 0, we add 3 on both sides.

In x + 5 = 0, we take away 5 on both sides.

So x = 0, x = 3 and x = −5.

Three answers.

4 Check

Let's check: put each answer into 2x³ + 4x² − 30x = 0.

First x = 0.

Now x = 3.

Now x = −5.

We get 0 every time.

So all three answers are right!

Let's look back.

First: get 0 on one side.

Then: factor completely.

Then: each factor is 0.

Last: check.