If a product is zero, at least one factor is zero. This is what turns a factored quadratic into two simple equations.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The right side must be zero
The property applies only to zero. From (x − 2)(x − 3) = 6 you cannot conclude anything about the factors.
Factoring a trinomial
For x² + bx + c, find two numbers multiplying to c and adding to b.
Difference of squares
a² − b² factors as (a + b)(a − b). Recognising it saves the whole search.
Common factor first
Always take out a common factor before anything else. It makes what remains far easier to factor.
It does not always work
Many quadratics do not factor over the integers. Those need the formula, which comes next.
The zero product property
If a product equals zero, at least one factor must be zero. That is why factoring solves equations, and it is why the equation must first be arranged with zero on one side.
Set it equal to zero first
Factoring x² + 5x = 6 and setting each factor to 6 is wrong, because the zero product property applies only to zero. Rearranging to x² + 5x − 6 = 0 is a mandatory first step.
Take out common factors first
For 2x² + 10x + 12, extract the 2 before factoring further. It keeps the remaining numbers small, and it is the step most often skipped.
Not everything factors
x² + x + 1 has no factorisation over the integers. That is not a failure of technique — the roots are irrational or complex. Knowing when to stop and use the formula saves considerable time.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Diego Factor and Solve
Diego solves x² − 5x + 6 = 0.
- Step 1
He needs two numbers multiplying to 6 and adding to −5.
Step 4: Your Turn
Practice makes it stick.
The Factors
Problem 1 of 2
(x − 4)(x + 2) = 0. What is the positive solution?
The Trinomial
Problem 2 of 2
x² − 5x + 6 = 0. What is the larger solution?
Factor and Solve
1 of 8
(x − 7)(x + 1) = 0. Larger solution?
2 of 8
x² − 9 = 0. Positive solution?
3 of 8
x² + 5x + 6 = 0. Larger solution?
4 of 8
x² − 7x = 0. Non-zero solution?
5 of 8
x² − 16 = 0. Positive solution?
6 of 8
x² − 4x + 4 = 0. What is the repeated solution?
7 of 8
Which expressions are a difference of squares?
8 of 8
x² − 8x + 15 = 0. Larger solution?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² − 6x + 8 = 0. What is the larger solution?
Question 2 of 2
Why must the equation equal zero before factoring helps?
What You Learned
- A product is zero only when a factor is zero.
- Rearrange to equal zero, then factor.
- Take out a common factor first and watch for a difference of squares.