Step by step
The question
Question: solve −x³ + 9x = 0.
The first part is negative.
We change every sign.
0 stays 0.
Now we have x³ − 9x = 0.
1 Factor completely
Common factor first.
Every term has x.
So we take out x.
So it is x(x² − 9) = 0.
Careful!
Don't divide both sides by x.
Then the answer x = 0 would be lost.
x² − 9 is one square take away another square.
x² is x squared.
9 is 3 squared.
A square minus a square is the plus bracket times the minus bracket.
So it is x(x + 3)(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 + 3 = 0, we take away 3 on both sides.
In x − 3 = 0, we add 3 on both sides.
So x = 0, x = −3 and x = 3.
Three answers.
3 Check
Let's check: put each answer into x³ − 9x = 0.
First x = 0.
Now x = −3.
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.