Solving Quadratics by Factoring- Complete Tutorial
What Is Factoring and Why Bother?
Factoring is the fastest way to solve a quadratic equation—when it works. Instead of plugging numbers into the quadratic formula, you reverse-engineer the equation into two simpler pieces. If the quadratic has clean integer solutions, factoring is usually 30 seconds of work instead of 2 minutes of formula memorization.
Most textbooks throw factoring at you around Algebra 1 or 2, then act surprised when students struggle. The truth is, the process is straightforward. You just need to know the steps.
What You Need Before Starting
- Comfortable with distributing values (FOIL in reverse)
- Know your multiplication tables cold
- Can identify factors of a number quickly
- Understand what a quadratic equation looks like: ax² + bx + c = 0
If you stumble on any of these, fix that gap first. Factoring builds on basics—you can't skip steps and expect it to click.
The Zero Product Property: The Only Rule That Matters
Everything in factoring comes down to one principle:
If A × B = 0, then A = 0 or B = 0.
That's it. When your factored equation gives you two binomials multiplied together equaling zero, you set each binomial equal to zero separately. One of them (or both) will give you a valid solution.
The Factoring Process, Step by Step
Step 1: Make Sure It's Standard Form
Your equation must equal zero. If you have x² + 5x + 6 = 12, subtract 12 from both sides first. You need x² + 5x - 6 = 0 before anything else.
Step 2: Find Two Numbers That Multiply to "c" and Add to "b"
For x² + bx + c, you're hunting for two numbers that:
- Multiply together to give you c
- Add together to give you b
Example: x² + 5x + 6
- c = 6, so look for factors of 6: 1×6 or 2×3
- b = 5, so which pair adds to 5? 2 + 3 = 5 ✓
Your numbers are 2 and 3.
Step 3: Write the Factored Form
Replace "b" with your two numbers, split across two binomials:
(x + 2)(x + 3) = 0
Step 4: Solve Each Binomial
Set each piece equal to zero and solve:
- x + 2 = 0 → x = -2
- x + 3 = 0 → x = -3
Done. Two solutions: x = -2 and x = -3.
Worked Examples
Example 1: Positive "c" with Positive "b"
Solve x² + 7x + 12 = 0
Find factors of 12 that add to 7: 3 and 4.
(x + 3)(x + 4) = 0
x = -3 or x = -4
Example 2: Negative "c"
Solve x² - 5x + 6 = 0
Factors of 6 that add to -5: -2 and -3.
(x - 2)(x - 3) = 0
x = 2 or x = 3
Example 3: Negative "c" with Negative "b"
Solve x² + x - 12 = 0
Factors of -12 that add to 1: 4 and -3.
(x + 4)(x - 3) = 0
x = -4 or x = 3
Example 4: Leading Coefficient Greater Than 1
Sometimes you get 2x² + 5x - 3 = 0. This is trickier.
You need factors of (2 × -3) = -6 that add to 5. That's 6 and -1.
Rewrite: 2x² + 6x - x - 3 = 0
Factor by grouping:
2x(x + 3) - 1(x + 3) = 0
(2x - 1)(x + 3) = 0
x = ½ or x = -3
When Factoring Won't Work
Not every quadratic factors nicely. Some have:
- Irrational solutions (√2, π, etc.)
- Complex/imaginary solutions
- Messy decimals or fractions that don't simplify
When that happens, use the quadratic formula. It's not a failure—factoring just isn't the right tool for those problems.
Quick Reference: Factoring vs. Quadratic Formula
| Method | Best When | Speed |
|---|---|---|
| Factoring | Clean integer factors exist | Fast (if you spot the factors) |
| Quadratic Formula | Factoring is messy or impossible | Reliable but slower |
| Graphing | You need visual confirmation | Depends on tools available |
Common Mistakes That Blow the Answer
- Forgetting to set the equation to zero first. Always check this before factoring.
- Finding numbers that multiply to "ac" instead of "c" when a ≠ 1. Be careful with leading coefficients.
- Dropping a negative sign. -2 and +2 look similar but give completely different answers.
- Not checking your work. Multiply your factored binomials back out. If you don't get the original equation, something's wrong.
Practice: Solve These on Your Own
Try these before checking answers:
- x² + 8x + 15 = 0
- x² - 9x + 20 = 0
- x² + 2x - 15 = 0
- 2x² - 7x + 3 = 0
Answers: 1) x = -3, -5 | 2) x = 4, 5 | 3) x = 3, -5 | 4) x = 3, ½
The Bottom Line
Factoring works when the numbers cooperate. When they don't, move on. The goal is solving the equation—not forcing a particular method. Know your options, know when each applies, and don't waste time on factoring when the quadratic formula is the cleaner path.