Worked examples
The question
Question: solve x³ − 9x = 0.
x3 − 9x = 0
The right side is 0 already.
x3 − 9x = 0
1 Factor completely
Common factor first.
Every term has x.
So we take out x.
common factor: x
So it is x(x² − 9) = 0.
x(x2 − 9) = 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 − 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.
a² − b² = (a + b)(a − b)
x2 = (x)², 9 = 3²
So it is x(x + 3)(x − 3) = 0.
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.
A × B × C = 0 → A = 0 or B = 0 or C = 0
Each factor with x gives a small equation.
x = 0 or x + 3 = 0 or x − 3 = 0
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.
x = 0 or x = −3 or x = 3
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.
x = 0: 03 − 9(0) = 0 + 0 = 0
Now x = −3.
x = −3: (−3)3 − 9(−3) = −27 + 27 = 0
Now x = 3.
x = 3: 33 − 9(3) = 27 − 27 = 0
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.
Answer x = 0 or x = −3 or x = 3
▶ Watch it solved
The question
Question: solve 2x³ − 18x = 0.
2x3 − 18x = 0
The right side is 0 already.
2x3 − 18x = 0
1 Factor completely
Common factor first.
Every term has x, and every number divides by 2.
So we take out 2x.
common factor: 2x
So it is 2x(x² − 9) = 0.
2x(x2 − 9) = 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 − 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.
a² − b² = (a + b)(a − b)
x2 = (x)², 9 = 3²
So it is 2x(x + 3)(x − 3) = 0.
2x(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.
A × B × C = 0 → A = 0 or B = 0 or C = 0
2 is a plain number.
It is never 0, so it does not make the product 0.
2 ≠ 0
Each factor with x gives a small equation.
x = 0 or x + 3 = 0 or x − 3 = 0
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.
x = 0 or x = −3 or x = 3
So x = 0, x = −3 and x = 3.
Three answers.
3 Check
Let's check: put each answer into 2x³ − 18x = 0.
First x = 0.
x = 0: 2(0)3 − 18(0) = 0 + 0 = 0
Now x = −3.
x = −3: 2(−3)3 − 18(−3) = −54 + 54 = 0
Now x = 3.
x = 3: 2(3)3 − 18(3) = 54 − 54 = 0
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.
Answer x = 0 or x = −3 or x = 3
▶ Watch it solved
The question
Question: solve 4x³ = 36x.
4x3 = 36x
1 Get 0 on one side
To factor, we want 0 on the right side.
So we take away 36x on both sides.
We do it on both sides, so they stay equal.
4x3 − 36x = 36x − 36x
Now we have 4x³ − 36x = 0.
4x3 − 36x = 0
2 Factor completely
Common factor first.
Every term has x, and every number divides by 4.
So we take out 4x.
common factor: 4x
So it is 4x(x² − 9) = 0.
4x(x2 − 9) = 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 − 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.
a² − b² = (a + b)(a − b)
x2 = (x)², 9 = 3²
So it is 4x(x + 3)(x − 3) = 0.
4x(x + 3)(x − 3) = 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
4 is a plain number.
It is never 0, so it does not make the product 0.
4 ≠ 0
Each factor with x gives a small equation.
x = 0 or x + 3 = 0 or x − 3 = 0
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.
x = 0 or x = −3 or x = 3
So x = 0, x = −3 and x = 3.
Three answers.
4 Check
Let's check: put each answer into 4x³ − 36x = 0.
First x = 0.
x = 0: 4(0)3 − 36(0) = 0 + 0 = 0
Now x = −3.
x = −3: 4(−3)3 − 36(−3) = −108 + 108 = 0
Now x = 3.
x = 3: 4(3)3 − 36(3) = 108 − 108 = 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 x = 0 or x = −3 or x = 3
▶ Watch it solved
