Quadratics on the SAT: lesson and practice
SAT quadratic questions test four things: solving (factor, take square roots, or use the quadratic formula), the vertex (the minimum or maximum point, at x = −b/(2a)), the sum and product of solutions (−b/a and c/a), and the number of solutions (from the discriminant b² − 4ac).
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Solving
| Form | Best method | Example |
|---|---|---|
| x² + bx + c = 0 | Factor | x² − 5x − 14 = (x − 7)(x + 2) → 7, −2 |
| (x − h)² = k | Square root, both signs | (x − 3)² = 16 → x = 7 or −1 |
| Anything else | Quadratic formula or Desmos | x = (−b ± √(b² − 4ac)) / 2a |
The vertex
In y = a(x − h)² + k, the vertex is (h, k). In y = ax² + bx + c, it is at x = −b/(2a); substitute to get the minimum or maximum value.
Two shortcuts
| Asked for | Formula |
|---|---|
| Sum of solutions | −b/a |
| Product of solutions | c/a |
| Number of real solutions | b² − 4ac: positive → 2, zero → 1, negative → 0 |
Common mistakes
Sample questions with answers
1. What are the solutions to x² − 5x − 14 = 0?
- x = −7 and x = 2
- x = 7 and x = −2 (correct)
- x = 14 and x = −1
- x = 5 and x = −14
(x − 7)(x + 2) = 0, so x = 7 or x = −2.
2. What is the vertex of y = 2(x + 3)² − 5?
- (3, −5)
- (−3, −5) (correct)
- (−3, 5)
- (−5, −3)
x + 3 = x − (−3), so h = −3 and k = −5.
3. How many real solutions does x² + 4x + 5 = 0 have?
- Zero (correct)
- Exactly one
- Exactly two
- Infinitely many
16 − 20 = −4 is negative, so there are no real solutions.
Frequently asked questions
How many quadratic questions are on the SAT?
Quadratics are a major part of Advanced Math, which makes up about 35% of SAT Math, roughly 13 to 15 questions.
Do I need to memorize the quadratic formula for the SAT?
It helps, but many SAT quadratics can be factored or solved with Desmos. Know the formula as a backup.
How do I find the minimum value of a quadratic?
Find the vertex x-value with −b/(2a), then substitute it into the function. The result is the minimum (if a > 0).
What is the discriminant used for?
b² − 4ac tells you how many real solutions a quadratic has: positive means two, zero means one, negative means none.