Factor x^3 = 25x Step by Step | Solving by factoring

Step by step

The question

Question: solve x³ = 25x.

x³ = 25x

1 Get 0 on one side

To factor, we want 0 on the right side.

So we take away 25x on both sides.

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

x³ − 25x = 25x − 25x

Now we have x³ − 25x = 0.

x³ − 25x = 0

2 Factor completely

Common factor first.

Every term has x.

So we take out x.

common factor: x

So it is x(x² − 25) = 0.

x(x² − 25) = 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.

x² − 25 is one square take away another square.

x² is x squared.

25 is 5 squared.

A square minus a square is the plus bracket times the minus bracket.

a² − b² = (a + b)(a − b)

x² = (x)², 25 = 5²

So it is x(x + 5)(x − 5) = 0.

x(x + 5)(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 + 5 = 0 or x − 5 = 0

x = 0 is done already.

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

In x − 5 = 0, we add 5 on both sides.

x = 0 or x = −5 or x = 5

So x = 0, x = −5 and x = 5.

Three answers.

4 Check

Let's check: put each answer into x³ − 25x = 0.

First x = 0.

x = 0: 0³ − 25(0) = 0 + 0 = 0

Now x = −5.

x = −5: (−5)³ − 25(−5) = −125 + 125 = 0

Now x = 5.

x = 5: 5³ − 25(5) = 125 − 125 = 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