Factor x^3 + 2x^2 = 15x Step by Step

Step by step

The question

Question: solve x³ + 2x² = 15x.

x³ + 2x² = 15x

1 Get 0 on one side

To factor, we want 0 on the right side.

So we take away 15x on both sides.

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

x³ + 2x² − 15x = 15x − 15x

Now we have x³ + 2x² − 15x = 0.

x³ + 2x² − 15x = 0

2 Factor completely

Common factor first.

Every term has x.

So we take out x.

common factor: x

So it is x(x² + 2x − 15) = 0.

x(x² + 2x − 15) = 0

Careful!

Don't divide both sides by x.

Then the answer x = 0 would be lost.

Careful: don't divide by x. We would lose x = 0.

Now the bracket.

We need two numbers.

They multiply to −15, and they add to 2.

? × ? = −15, ? + ? = 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.

× is − → one +, one −; bigger one +

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.

1 × 15: 15 − 1 = 14

3 × 5: 5 − 3 = 2 ✓

So it is x(x − 3)(x + 5) = 0.

x(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.

A × B × C = 0 → A = 0 or B = 0 or C = 0

Each factor with x gives a small equation.

x = 0 or x − 3 = 0 or x + 5 = 0

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.

x = 0 or x = 3 or x = −5

So x = 0, x = 3 and x = −5.

Three answers.

4 Check

Let's check: put each answer into x³ + 2x² − 15x = 0.

First x = 0.

x = 0: 0³ + 2(0)² − 15(0) = 0 + 0 + 0 = 0

Now x = 3.

x = 3: 3³ + 2(3)² − 15(3) = 27 + 18 − 45

= 0

Now x = −5.

x = −5: (−5)³ + 2(−5)² − 15(−5)

= −125 + 50 + 75 = 0

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.

Answer