Quadratic equations are one of the most frequently tested topics in SAT Advanced Math — and one of the most learnable. The SAT tests a predictable set of quadratic patterns, and once you've seen each one enough times, they become almost automatic. This guide covers every method you need and the exact patterns the SAT uses.
What Is a Quadratic Equation?
A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0. The solutions (also called roots or zeros) are the x-values where the parabola crosses the x-axis.
The SAT tests quadratics in several forms — standard form, factored form, and vertex form — and you need to be comfortable converting between them.
The Three Solution Methods
Method 1: Factoring
The fastest method when it works. Find two numbers that multiply to ac and add to b.
Best for: integer coefficients, "nice" numbers. The SAT heavily favours factorable quadratics.
Method 2: Quadratic Formula
Works for every quadratic, even when factoring is difficult or impossible.
Best for: non-factorable quadratics, or when the question asks about the discriminant (b² − 4ac).
Method 3: Completing the Square
Rewrites the equation into vertex form. Less commonly needed for solving, but essential for understanding parabola properties.
Best for: finding the vertex, converting to vertex form, or when the SAT asks about the minimum/maximum value.
The Discriminant — SAT's Favourite
The discriminant is b² − 4ac, and the SAT loves questions about it:
- b² − 4ac > 0 → Two distinct real solutions (parabola crosses x-axis twice)
- b² − 4ac = 0 → One real solution (parabola is tangent to x-axis)
- b² − 4ac < 0 → No real solutions (parabola doesn't cross x-axis)
The 5 SAT Quadratic Patterns
For ax² + bx + c = 0, the sum of roots = −b/a and the product of roots = c/a.
SAT twist: "The product of the solutions of 2x² − 8x + 6 = 0 is..." → c/a = 6/2 = 3.
a² − b² = (a + b)(a − b). Very commonly tested.
SAT example: x² − 49 = (x + 7)(x − 7). No middle term means difference of squares.
x² + 2bx + b² = (x + b)². The SAT uses these in completing-the-square questions.
SAT example: x² + 10x + 25 = (x + 5)² → vertex at x = −5.
The SAT gives the same quadratic in two forms and asks you to match coefficients.
SAT example: If x² − 6x + k = (x − 3)² − 4, what is k? Expand right side: x² − 6x + 9 − 4 = x² − 6x + 5. So k = 5.
Word problems where you set up and solve a quadratic from a real-world scenario.
SAT example: A ball is thrown upward. Height h = −16t² + 80t. When does it hit the ground? Set h = 0: t(−16t + 80) = 0 → t = 0 or t = 5.
Vertex Form
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola.
- If a > 0, the parabola opens upward (minimum at vertex)
- If a < 0, the parabola opens downward (maximum at vertex)
- The axis of symmetry is x = h
The SAT frequently asks: "What is the minimum value of f(x) = 2(x − 3)² + 5?" Answer: 5 (when x = 3, the squared term is 0, leaving just the k value).
Top 5 Tips for SAT Quadratics
- Try factoring first — the SAT deliberately makes most quadratics factorable. If you don't see factors within 30 seconds, switch to the quadratic formula.
- Memorise the quadratic formula — you will need it. Say it aloud repeatedly until it's automatic: "negative b, plus or minus square root of b-squared minus 4ac, all over 2a."
- Watch for the zero product property — if (x − 2)(x + 5) = 0, then x = 2 or x = −5. The SAT sometimes gives factored form directly.
- The discriminant is your friend — any question about "how many solutions" or "what value of k makes the equation have one solution" is a discriminant question.
- Use Desmos on the real test — for hard contextual quadratic questions, graphing the parabola in Desmos and reading the zeros is often faster than algebra.
Practice SAT quadratic equations now
Easy, Medium, and Hard quadratic questions with full step-by-step explanations.