Factoring Quadratics — SAT Math Explained
Rewriting a quadratic expression ax² + bx + c as a product of two binomials (x + p)(x + q), using patterns and number relationships.
The Core Idea
Factoring is the reverse of expanding (FOIL). You're asking: 'What two binomials, when multiplied, give me this trinomial?' The key is finding two numbers that multiply to give ac and add to give b.
Key Vocabulary
To write as a product; to break apart into multiplied pieces
A polynomial with exactly two terms
A polynomial with exactly three terms
The largest factor that divides all terms — always factor this out first
If A × B = 0, then A = 0 or B = 0 (the key to solving after factoring)
Factoring Strategies
Always check for and factor out the Greatest Common Factor before anything else
Find two numbers that multiply to c and add to b: x² + bx + c = (x + p)(x + q)
Multiply a × c, find factors of this product that add to b, split the middle term, factor by grouping
a² - b² = (a + b)(a - b) — recognize this pattern
a² + 2ab + b² = (a + b)² — recognize the perfect square pattern
Zero Product Property
Once factored into (x + p)(x + q) = 0, set each factor to zero and solve: x + p = 0 → x = -p; x + q = 0 → x = -q
Checking Work
Expand your factored form using FOIL or distribution — you should get back the original expression
Common Errors to Avoid
Forgetting to factor out the GCF first
Sign errors when finding p and q (especially with negative middle terms)
Not applying the Zero Product Property after factoring — factoring alone doesn't solve the equation
Practice: Factoring Quadratics
5 SAT-style questions. Select your answer and get an instant explanation.
Factor: x² + 5x + 6
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