Quadratic Equations · Sub-skill drill
Factoring Quadratic Expressions
Factoring is usually the fastest method for solving SAT quadratics where the leading coefficient is 1 and the roots are integers. Look for two integers whose product is the constant term and whose sum is the linear coefficient; these are your factors. The roots are then the negatives of those factors. For quadratics with a leading coefficient other than 1, use the AC method or grouping. The College Board designs most SAT quadratics to factor cleanly, so reach for the quadratic formula only when factoring fails.
How this sub-skill is tested on the SAT
Factoring is usually the fastest method for solving SAT quadratics where the leading coefficient is 1 and the roots are integers. Look for two integers whose product is the constant term and whose sum is the linear coefficient; these are your factors. The roots are then the negatives of those factors. For quadratics with a leading coefficient other than 1, use the AC method or grouping. The College Board designs most SAT quadratics to factor cleanly, so reach for the quadratic formula only when factoring fails.
This sub-skill sits inside the broader Quadratic Equations topic, which is part of the College Board's Passport to Advanced Math content domain. Passport to Advanced Math is the SAT's bridge to the kind of algebraic manipulation you will see in a college precalculus or calculus course. The questions are not about memorizing identities — they are about fluency with structure. Can you factor a quadratic by inspection? Can you read the vertex of a parabola off its standard form? Can you simplify a rational expression without losing a domain restriction? Can you translate between f(x), a graph, and a table of values without panicking? Most students who plateau between 650 and 720 plateau here, because the section rewards algebraic intuit
Practice questions in this drill set
Below are 7 practice questions targeting this exact sub-skill, ordered from easier to harder. Each question is tagged with its target score band so you can focus on questions that match the band you are working out of. Worked solutions are open by default — read each one even if you got the question right, because the way the solution is structured often reveals a faster path than the one you used.
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What are the solutions to the equation x^2 - 11x + 30 = 0?
- A x = -5 and x = -6
- B x = 30 only
- C x = 5 and x = 6
- D x = 6 and x = 7
Worked solution
Answer: C — x = 5 and x = 6
Factor: x^2 + (-11)x + (30) = (x - 5)(x - 6) = 0. Setting each factor to zero gives x = 5 or x = 6. Verify by Vieta: sum of roots = 5 + 6 = 11 = -b ✓; product of roots = 5 × 6 = 30 = c ✓.
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What are the solutions to the equation x^2 + 5x + 6 = 0?
- A x = 6 only
- B x = 3 and x = 2
- C x = -3 and x = -2
- D x = -2 and x = -1
Worked solution
Answer: C — x = -3 and x = -2
Factor: x^2 + (5)x + (6) = (x - -3)(x - -2) = 0. Setting each factor to zero gives x = -3 or x = -2. Verify by Vieta: sum of roots = -3 + -2 = -5 = -b ✓; product of roots = -3 × -2 = 6 = c ✓.
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What are the solutions to the equation x^2 + 9x + 20 = 0?
- A x = 5 and x = 4
- B x = 20 only
- C x = -4 and x = -3
- D x = -5 and x = -4
Worked solution
Answer: D — x = -5 and x = -4
Factor: x^2 + (9)x + (20) = (x - -5)(x - -4) = 0. Setting each factor to zero gives x = -5 or x = -4. Verify by Vieta: sum of roots = -5 + -4 = -9 = -b ✓; product of roots = -5 × -4 = 20 = c ✓.
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What are the solutions to the equation x^2 - 3x - 18 = 0?
- A x = 7 and x = -2
- B x = -18 only
- C x = -6 and x = 3
- D x = 6 and x = -3
Worked solution
Answer: D — x = 6 and x = -3
Factor: x^2 + (-3)x + (-18) = (x - 6)(x - -3) = 0. Setting each factor to zero gives x = 6 or x = -3. Verify by Vieta: sum of roots = 6 + -3 = 3 = -b ✓; product of roots = 6 × -3 = -18 = c ✓.
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What are the solutions to the equation x^2 + 4x - 12 = 0?
- A x = -6 and x = 2
- B x = 6 and x = -2
- C x = -12 only
- D x = -5 and x = 3
Worked solution
Answer: A — x = -6 and x = 2
Factor: x^2 + (4)x + (-12) = (x - -6)(x - 2) = 0. Setting each factor to zero gives x = -6 or x = 2. Verify by Vieta: sum of roots = -6 + 2 = -4 = -b ✓; product of roots = -6 × 2 = -12 = c ✓.
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What are the solutions to the equation x^2 + 1x - 20 = 0?
- A x = 4 and x = -5
- B x = -4 and x = 5
- C x = 5 and x = -4
- D x = -20 only
Worked solution
Answer: A — x = 4 and x = -5
Factor: x^2 + (1)x + (-20) = (x - 4)(x - -5) = 0. Setting each factor to zero gives x = 4 or x = -5. Verify by Vieta: sum of roots = 4 + -5 = -1 = -b ✓; product of roots = 4 × -5 = -20 = c ✓.
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What are the solutions to the equation x^2 + 9x + 20 = 0?
- A x = 5 and x = 4
- B x = 20 only
- C x = -4 and x = -3
- D x = -5 and x = -4
Worked solution
Answer: D — x = -5 and x = -4
Factor: x^2 + (9)x + (20) = (x - -5)(x - -4) = 0. Setting each factor to zero gives x = -5 or x = -4. Verify by Vieta: sum of roots = -5 + -4 = -9 = -b ✓; product of roots = -5 × -4 = 20 = c ✓.
Why this band assignment matters
Every question in this drill is tagged with a target score band — 400–500, 500–600, 600–700, or 700–800 — based on its difficulty and the patterns the College Board uses for questions at each level. If you are aiming to break out of a 580 plateau, the 600–700 questions in this drill are your highest-leverage practice. If you are chasing 750+, the 700–800 questions here are the ones that separate the top 10% of test takers from everyone else.
Use the band tags to filter your work. If you can confidently solve every 400–500 and 500–600 question without notes, move to the 600–700 set. If those land cleanly, the 700–800 set is your final boss. The worked solutions in this drill are written so that even the hardest questions become learnable patterns once you have seen the structure of the solve a few times.