Step by step
The question
Question: solve x³ − 6x² + 9x = 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 − 6x + 9) = 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 9, and they add to −6.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 9, smallest first.
We add each pair.
We need 6.
3 and 3 give 6.
Yes!
With the signs: −3 and −3.
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 add 3 on both sides.
So x = 0 and x = 3.
Two answers.
3 Check
Let's check: put each answer into x³ − 6x² + 9x = 0.
First x = 0.
Now x = 3.
We get 0 every time.
So both answers are right!
Let's look back.
First: factor completely.
Then: each factor is 0.
Last: check.