Solving Quadratic Equations by Factoring
When a quadratic factors nicely, factoring is the fastest way to solve it. The engine is the zero product property: if a product is zero, at least one factor must be zero.
Set the equation to zero, factor, then solve each factor separately. Two factors usually give two solutions.
The zero product property
If , then or . A product equals zero only when one of its factors is zero.
So after factoring, split into separate simple equations — one per factor — and solve each for .
Set to zero first
The property only works when the product equals zero, so move every term to one side first. Then factor the resulting expression.
For , factor into and set each factor to zero.
Worked examples
Example 1: factor and solve
Solve by factoring.
Answer: or
Example 2: setting to zero first
Rewrite so factoring applies.
Answer:
Try one yourself
Common questions
What is the zero product property?
If a product of factors equals zero, at least one factor must be zero. That lets you turn a factored quadratic into simple linear equations.
Why must the equation equal zero first?
The property only applies to a product equal to zero. If the right side is not zero, the logic fails, so move all terms to one side.
What if the quadratic does not factor?
Use the quadratic formula or completing the square. Factoring is fastest only when clean integer factors exist.
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