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.
A quadratic is an equation with an x² term as its highest power — something like x² - 7x + 10 = 0. Factoring means rewriting that expression as a product of two binomials (two-term expressions), like (x - 2)(x - 5). Why bother? Because of one golden rule:
Zero Product Property: if two things multiply to 0, then at least one of them must be 0. So if (x - 2)(x - 5) = 0, then either x - 2 = 0 or x - 5 = 0, giving x = 2 or x = 5. Those are the solutions (also called roots or zeros).
Here's the factoring recipe for x² + bx + c (when the number in front of x² is 1):
Find two numbers that multiply to c and add to b.
Drop those two numbers into (x + ___)(x + ___).
Example: x² - 7x + 10. I need two numbers that multiply to +10 and add to -7. Those are -2 and -5 (since -2 × -5 = 10 and -2 + -5 = -7). So it factors to (x - 2)(x - 5).
Sign shortcuts save time:
If c is positive, both numbers share the same sign (both match the sign of b).
If c is negative, the numbers have opposite signs.
Two facts worth memorizing (called Vieta's formulas) for x² + bx + c = 0:
Sum of the solutions = -b
Product of the solutions = c
So for x² - 8x + 7 = 0, the solutions add up to 8 — no factoring needed if that's all they ask!
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?
What is the positive solution to the equation x² - 7x + 10 = 0?
If x² - 8x + 7 = 0, what is the sum of the solutions?
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.
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.