Algebra
Quadratic equations and functions
Factoring quadratic equations
20 practice questions
2 video lessons
Theory + worked examples
Theory
Factoring solves a quadratic using the zero product property: if a product is \(0\), one factor is \(0\).
- Write the quadratic in standard form (\(=0\)).
- Factor it.
- Set each factor equal to zero.
- Solve each for the roots.
Only works when the equation equals zero and factors nicely.
Roots at \(x=-2\) and \(x=-3\).
Factoring to solve.
Zero product property:
\[(x-p)(x-q)=0\ \Rightarrow\ x=p\ \text{or}\ x=q\]
Each factor set to zero gives a root.
How to factor and solve
- Set the equation to \(=0\).
- Factor the quadratic.
- Set each factor equal to zero.
- Solve for each root.
Example 1 — Factor and solve
Solve \(x^2+5x+6=0\).
Solution
Factor, then use the zero product property.
| \((x+2)(x+3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(-2,\ -3\) |
Example 2 — Difference of squares
Solve \(x^2-9=0\).
Solution
Factor the difference of squares.
| \((x-3)(x+3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(3,\ -3\) |
Example 3 — With a GCF
Solve \(2x^2+6x=0\).
Solution
Factor out \(2x\).
| \(2x(x+3)\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(0,\ -3\) |
Example 4 — Zero product property
Why does factoring solve a quadratic?
Solution
If a product is \(0\), at least one factor is \(0\) — the zero product property.
Common pitfalls
Set the equation to zero first, then factor.
Set EACH factor to zero, not the product.
Don't divide by \(x\) — you'd lose the root \(x=0\).
Frequently asked questions
How does factoring solve a quadratic?
By the zero product property: set each factor to zero.
Solve \((x-4)(x+1)=0\).
\(x=4\) or \(x=-1\).
Do you set the equation to zero first?
Yes, before factoring.
Can you always factor a quadratic?
No — some need the quadratic formula.
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