Same Math › Algebra 1 · grades 8–10 › The zero product property
What is the zero product property?
If two numbers multiply to zero, one of them has to be zero. That's the zero product property, and it's why factoring solves equations. (x − 3)(x + 5) = 0 means x − 3 = 0 or x + 5 = 0, so x = 3 or x = −5. Kids set each factor equal to zero and solve two small equations.
- Factored and equal to 0: the only time this works.
- Each factor = 0 gives one answer.
- The answers are called solutions, roots or zeros.
Why schools teach it
It's the reason factoring matters
Algebra 1 students solve quadratic equations by factoring (HSA.REI.B.4b) and connect the answers to where a graph crosses the x-axis (HSA.SSE.B.3a).
Kids often factor correctly and then stop, or plug in numbers at random. Naming the property makes the last step automatic: set each factor to zero.
At the kitchen table
What to say, and what to skip
Say this
Is one side zero?
If not, move everything over first.Is it factored yet?
Set each factor equal to zero. What do you get?
Plug your answers back in. Do you get zero?
Skip this
- Dividing both sides by x.That loses the answer x = 0 in x² = 5x.
- Using the property when the right side isn't 0.(x − 1)(x + 2) = 4 must be rearranged first.
- Writing x = −3 for the factor (x − 3).x − 3 = 0 gives x = 3. The sign flips.
See it in action
Solving x² + x = 12
Get zero on one side
x² + x − 12 = 0Factor
(x + 4)(x − 3) = 0Set each factor to zero
x + 4 = 0 or x − 3 = 0Solve both
x = −4 or x = 3
Two answers is normal for a quadratic.
Free practice sheet
Make a zero product sheet
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Parents also ask
Why are there two answers?
A quadratic can cross the x-axis twice. Each crossing is a solution.
What about 2x(x − 7) = 0?
Three factors: 2, x and x − 7. The 2 can't be zero, so x = 0 or x = 7.
What if a factor is 2x + 3?
Solve 2x + 3 = 0 to get x = −3/2. Fractions are normal here.
What are roots and zeros?
Other names for the solutions. On a graph they're the x-intercepts.