Step by Step
1
Get one side to equal zero
Rearrange the equation so everything is on one side and it equals zero on the other.
2
Factor the expression
Factor the non-zero side into two (or more) factors.
3
Set each factor equal to zero
Since the product of the factors equals zero, at least one of the individual factors must itself equal zero.
4
Solve each mini-equation
Solve each of the resulting simple equations separately — each gives you one solution to the original equation.
Applied Walkthrough
1
Solve x² + x − 6 = 0. The equation already equals zero, so factor the left side: (x + 3)(x − 2) = 0.
2
Apply the zero product property: since (x+3)(x−2) = 0, either (x+3) = 0 or (x−2) = 0.
3
Solve each: x + 3 = 0 gives x = −3. x − 2 = 0 gives x = 2.
4
The solutions are x = −3 and x = 2 — both values make the original equation true.
Exam Application
Exams test whether you understand why factoring is only useful for solving an equation once one side equals zero, and whether you can correctly set each factor to zero and solve both resulting mini-equations.
⚠ Common Trap
The most common trap is trying to apply the zero product property before the equation actually equals zero — for example, factoring x² + x = 6 directly without first rearranging it to x² + x − 6 = 0.
✓ Quick Self-Check
1. What does the zero product property state?
If AB = 0, then A = 0 or B = 0.
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2. What must be true about an equation before you can apply this property?
One side of the equation must equal zero.
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3. Solve x² + x − 6 = 0.
x = −3 or x = 2.
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4. Why can't you apply the zero product property to x² + x = 6 directly?
Because the equation doesn't yet equal zero — you must rearrange it first.
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5. What two methods build directly on the zero product property?
Factoring-based solving, which connects to both FOIL (for expanding) and the quadratic formula (as an alternative when factoring is hard).
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