Factoring Quadratic Expressions
Key patterns, steps, and shortcuts for factoring quadratics on the PSAT.
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Study Smart Tips for Factoring Quadratic Expressions
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Questions Covered in This Set
12 cards to master
What is the very first step in any factoring problem?
Pull out the greatest common factor (GCF) from every term — numbers and variables — before trying any pattern.
Difference of squares formula
a² − b² = (a + b)(a − b) — two perfect-square terms separated by a minus sign.
Does x² + 49 factor over the real numbers?
No. A sum of squares does not factor over the reals; only a difference of squares does.
How do you factor x² + bx + c (leading coefficient 1)?
Find two numbers that multiply to c and add to b; then it factors as (x + m)(x + n).
Factor x² + 2x − 15
Product −15, sum 2 → 5 and −3, so (x + 5)(x − 3).
Sign shortcut for trinomials
If c > 0, both numbers share b's sign. If c < 0, the numbers have opposite signs and the larger takes b's sign.
Explain the ac method for ax² + bx + c
Find two numbers multiplying to a·c and adding to b, split the middle term, then factor by grouping.
Factor 6x² + 11x − 10 using the ac method
ac = −60, sum 11 → 15 and −4: 6x² + 15x − 4x − 10 = 3x(2x+5) − 2(2x+5) = (2x+5)(3x−2).
If x² − y² = 40 and x + y = 8, what is x − y?
(x+y)(x−y) = 40, so 8(x−y) = 40 and x − y = 5.
Why is factoring useful for solving equations?
A product equal to zero gives instant solutions: if (x−3)(x+5)=0, then x = 3 or x = −5.
Factor x³ − 4x completely
x(x² − 4) = x(x + 2)(x − 2).
What habit should you use after factoring?
FOIL/multiply back mentally to verify — it takes five seconds and catches sign errors.
