Step by step
The question
Question: solve 3x³ + 3x² − 18x = 0.
The right side is 0 already.
1 Factor completely
Common factor first.
Every term has x, and every number divides by 3.
So we take out 3x.
So it is 3x(x² + x − 6) = 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 −6, and they add to 1.
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.
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
So it is 3x(x − 2)(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.
3 is a plain number.
It is never 0, so it does not make the product 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 + 3 = 0, we take away 3 on both sides.
So x = 0, x = 2 and x = −3.
Three answers.
3 Check
Let's check: put each answer into 3x³ + 3x² − 18x = 0.
First x = 0.
Now x = 2.
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.