Factoring Quadratics
Quadratic equations show up all over the Math section, and the fastest way to crack most of them is factoring — turning a messy equation into two simple ones you can solve in seconds.
| Need | Value | Pair |
|---|---|---|
| Multiply to c | +10 | -2 and -5 |
| Add to b | -7 | -2 + (-5) = -7 |
| Factored form | (x-2)(x-5) | x = 2, x = 5 |
Find a pair that multiplies to c and adds to b, then read off the solutions.
The solutions of x² - 7x + 10 = 0 are the x-intercepts: x = 2 and x = 5.
If the x² term has a coefficient other than 1 (like 2x²), factoring is trickier — but on the test you can often fall back on the quadratic formula or just plug in answer choices. Master the x²-coefficient-of-1 case first, because it covers most factoring questions you'll see.
What are the solutions to x² + -12x + 32 = 0?
Worked examples
What are the solutions to x² + -12x + 32 = 0?
What is the positive solution to the equation x² - 7x + 10 = 0?
If x² - 8x + 7 = 0, what is the sum of the solutions?
Common pitfalls
If you factor to (x - 5), the solution is x = +5, not -5. Set each factor equal to zero and solve — don't just copy the number inside the parentheses.
The two numbers must multiply to c AND add to b — both at once. Students often find a pair that multiplies correctly but forget to check the sum. Always verify both.
When c is negative, the two numbers have opposite signs, so one solution is positive and one is negative. Don't make both the same sign — check that your pair actually multiplies to a negative.
The Zero Product Property only works when one side is 0. If you see x² - 7x = -10, move everything to one side first: x² - 7x + 10 = 0 before factoring.
Key takeaways
To factor x² + bx + c, find two numbers that multiply to c and add to b.
Zero Product Property: if (x - p)(x - q) = 0, then x = p or x = q — flip the sign inside each parenthesis.
If c is positive, the two numbers share b's sign; if c is negative, they have opposite signs.
Sum of solutions = -b and product of solutions = c (Vieta's formulas) — use these to skip factoring.
Always set the equation equal to 0 before applying the Zero Product Property.
Try it yourself
5 practice questions on Factoring Quadratics, drawn from the question bank. The tutor is one click away if you get stuck.