Step by step
The question
Question: solve x³ = 7x² − 10x.
1 Get 0 on one side
To factor, we want 0 on the right side.
So we take away 7x² − 10x on both sides.
We do it on both sides, so they stay equal.
Now we have x³ − 7x² + 10x = 0.
2 Factor completely
Common factor first.
Every term has x.
So we take out x.
So it is x(x² − 7x + 10) = 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 10, and they add to −7.
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 10, smallest first.
We add each pair.
We need 7.
2 and 5 give 7.
Yes!
With the signs: −2 and −5.
So it is x(x − 2)(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.
Each factor with x gives a small equation.
x = 0 is done already.
In x − 2 = 0, we add 2 on both sides.
In x − 5 = 0, we add 5 on both sides.
So x = 0, x = 2 and x = 5.
Three answers.
4 Check
Let's check: put each answer into x³ − 7x² + 10x = 0.
First x = 0.
Now x = 2.
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.