Direct Answer
To solve a quadratic equation by factoring: (1) move all terms to one side so the equation reads ax² + bx + c = 0; (2) find two numbers that multiply to a·c and add to b; (3) use them to write the equation in factored form, e.g. (x − 2)(x − 3) = 0; (4) set each factor equal to zero using the zero product property; (5) solve the two mini-equations. For x² − 5x + 6 = 0, the factors are (x − 2)(x − 3), giving x = 2 and x = 3.
Why Factoring Is the Fastest Method (When It Works)
Most students reach straight for the quadratic formula — and it works every time, which is exactly why it’s a crutch. When an equation factors neatly, factoring is dramatically faster: x² − 5x + 6 = 0 solves in about ten seconds once you spot the pair. No square roots, no ± branches, no arithmetic with big numbers. Exam setters know this; the first questions on most algebra tests are chosen to factor.
The 5 Steps to Solve by Factoring
Step 1: Write the equation in standard form. Everything goes to one side: ax² + bx + c = 0. If you start with x² − 5x = −6, add 6 to both sides first.
Step 2: Find the factor pair. Find two numbers that multiply to a·c and add to b. For x² − 5x + 6 = 0: a·c = 1·6 = 6, b = −5. The pair is −2 and −3: (−2)(−3) = 6 and (−2) + (−3) = −5.
Step 3: Write the factored form. For a = 1: (x − 2)(x − 3) = 0. Expand mentally to check: x² − 3x − 2x + 6 = x² − 5x + 6.
Step 4: Apply the zero product property. (x − 2)(x − 3) = 0 means x − 2 = 0 or x − 3 = 0.
Step 5: Solve the mini-equations. x = 2 or x = 3. Substitute both back to verify.
Worked Example 1: x² − 5x + 6 = 0 (the 10-second solve)
(x − 2)(x − 3) = 0 → x = 2, x = 3. Verification: 2² − 5(2) + 6 = 0; 3² − 5(3) + 6 = 0. This is the exact example in Calcrio’s free quadratic equation calculator.
Worked Example 2: When a ≠ 1 — Factor by Grouping
Solve 2x² − 7x + 3 = 0. a·c = 6, b = −7 → pair −6 and −1. 1. Split: 2x² − 6x − x + 3 = 0. 2. Group: 2x(x − 3) − 1(x − 3) = 0. 3. (x − 3)(2x − 1) = 0. 4. x = 3; x = 1/2.
Worked Example 3: Difference of Squares
Solve x² − 49 = 0: (x − 7)(x + 7) = 0 → x = 7 or x = −7.
When Factoring Fails — Switch to the Formula
x² + x + 1 = 0 has no integer pair (discriminant −3) — no real roots to factor toward. Rule of thumb: give factoring ~30 seconds, then switch to the formula. The discriminant tells you which kind you’re dealing with. Calcrio’s quadratic equation calculator shows both modes.
Why Do Students Get Factoring Wrong?
Sign errors: hunt for “multiply to 6, add to −5” — both numbers negative. Forgetting step 1: move terms first, factor second. Giving up on a ≠ 1: use split-the-middle-term. Trusting unexpanded factors: always expand mentally to check.
Practice Set: Five Quadratics to Factor
- x² + 7x + 12 = 0 → (x + 3)(x + 4) = 0 → x = −3, x = −4.
- x² − 9x + 20 = 0 → (x − 4)(x − 5) = 0 → x = 4, x = 5.
- x² + 2x − 15 = 0 → (x + 5)(x − 3) = 0 → x = −5, x = 3.
- x² − 11x + 30 = 0 → (x − 5)(x − 6) = 0 → x = 5, x = 6.
- 3x² + 10x + 8 = 0 → (3x + 4)(x + 2) = 0 → x = −4/3, x = −2.
How Long Does Factoring Take to Learn?
Drill: pick target product and sum (“multiply to 12, add to −7” → −3 and −4), write ten pairs a day. Within two weeks the common triples fire instantly.
FAQ
How do you know if a quadratic can be factored? Compute D = b² − 4ac. If D is a perfect square, it factors over rational numbers.
What is the zero product property? If a·b = 0, then a = 0 or b = 0.
Can every quadratic be solved by factoring? No — only rational-root cases. The formula covers the rest.
What if there’s no x term or constant? Factor out what’s common: x² − 5x = 0 → x = 0 or 5; x² − 49 = 0 → ±7.
Do I need to memorize factor pairs? No — practice builds speed.
Next: What Is the Discriminant? b²−4ac Explained with Examples