Factoring Quadratics — SAT Math Guide

Q: What You Need to Know

• Standard form: ax² + bx + c where a, b, and c are constants • Factor pairs must multiply to give "ac" and add to give "b" • Perfect square trinomials: a² + 2ab + b² = (a + b)² • Difference of squares: a² - b² = (a + b)(a - b) • Always check your factors by expanding back to the original expression

Factoring quadratics SAT problems appear regularly on the Digital SAT and can make or break your advanced math score. This technique involves breaking down quadratic expressions like x² + 5x + 6 into their component factors like (x + 2)(x + 3). The College Board includes factoring quadratics in roughly 2-3 questions per SAT math section, making it essential for your SAT math prep. With the right approach, these questions become straightforward point-scorers.

What You Need to Know

Standard form: ax² + bx + c where a, b, and c are constants

Factor pairs must multiply to give "ac" and add to give "b"

Perfect square trinomials: a² + 2ab + b² = (a + b)²

Difference of squares: a² - b² = (a + b)(a - b)

Always check your factors by expanding back to the original expression

Some quadratics cannot be factored using integers (irreducible)

📐 KEY FORMULA: ax² + bx + c = (px + q)(rx + s) where pr = a, qs = c, ps + qr = b

💡 PRO TIP: On the Digital SAT, if factoring seems too complex, try substituting the answer choices back into the original equation.

How to Solve Factoring Quadratics on the SAT

Example Question 1 — Medium Difficulty

Which of the following is equivalent to x² - 7x + 12?

A) (x - 3)(x - 4)

B) (x + 3)(x + 4)

C) (x - 2)(x - 6)

D) (x + 2)(x + 6)

Solution:

Step 1: Identify a = 1, b = -7, c = 12

Step 2: Find two numbers that multiply to 12 and add to -7

Step 3: The numbers are -3 and -4 because (-3) × (-4) = 12 and (-3) + (-4) = -7

✅Answer: A — The factored form is (x - 3)(x - 4).

Example Question 2 — Hard Difficulty

If 2x² - 8x + 6 = 2(x - a)(x - b) where a and b are constants, what is the value of a + b?

A) 2

B) 3

C) 4

D) 6

Solution:

Step 1: Factor out the common factor: 2x² - 8x + 6 = 2(x² - 4x + 3)

Step 2: Factor the quadratic inside: x² - 4x + 3 = (x - 1)(x - 3)

Step 3: So 2x² - 8x + 6 = 2(x - 1)(x - 3), meaning a = 1 and b = 3

Step 4: Therefore a + b = 1 + 3 = 4

✅Answer: C — After factoring, a + b equals 4.

Common SAT Math Mistakes to Avoid

❌Mistake: Forgetting to factor out the greatest common factor first

✅Fix: Always check if all terms share a common factor before attempting other factoring methods

❌Mistake: Getting the signs wrong when factoring

✅Fix: Pay careful attention to whether you need factors that add or subtract to get the middle term

❌Mistake: Not checking your answer by expanding

✅Fix: Always multiply your factors back out to verify they equal the original expression

❌Mistake: Assuming all quadratics can be factored with integers

✅Fix: If you can't find integer factors quickly, consider that the quadratic might be irreducible or require the quadratic formula

Practice Question — Try It Yourself

What is the factored form of 3x² + 11x + 6?

A) (3x + 2)(x + 3)

B) (3x + 3)(x + 2)

C) (3x + 1)(x + 6)

D) (3x + 6)(x + 1)

Show Answer

Answer: A — We need factors of 3 × 6 = 18 that add to 11. Those are 9 and 2, giving us (3x + 2)(x + 3).

Key Takeaways for the SAT

Master the basic factoring patterns before test day — they save valuable time on the SAT math section

When stuck, try working backwards by expanding the answer choices

Always factor out common factors first to simplify your work

SAT math factoring quadratics often connects to solving equations or finding zeros

Practice recognizing special cases like perfect squares and difference of squares for quick solutions