Quadratics are one of the most heavily tested topics on the Digital SAT Math section. Expect 8 to 12 questions involving quadratic functions, equations, or expressions across both modules. They show up as factoring problems, vertex form conversions, discriminant questions, and word problems. Mastering them unlocks a real score boost. This guide covers everything about SAT quadratic equations and functions.
What Is a Quadratic on the SAT?
A quadratic is any function or equation where x is raised to the power of 2 (but no higher). General form: y = ax^2 + bx + c, with a not equal to 0. The graph is always a parabola. Quadratics open upward when a is positive and downward when a is negative. SAT questions test your ability to work with quadratics in standard form, factored form, and vertex form.
The Three Forms of Quadratics
Standard Form: y = ax^2 + bx + c
Useful for reading the y-intercept (c) and the leading coefficient (a). The y-intercept is always (0, c).
Factored Form: y = a(x – r1)(x – r2)
Useful for reading the roots (r1, r2). These are where the parabola crosses the x-axis.
Vertex Form: y = a(x – h)^2 + k
Useful for reading the vertex (h, k). The vertex is the maximum or minimum point of the parabola.
Key Techniques You Need
Factoring
For a quadratic x^2 + bx + c, find two numbers that multiply to c and add to b. Example: x^2 + 5x + 6 factors to (x + 2)(x + 3). Practice factoring until it feels automatic. Many SAT questions hinge on quick factoring.
The Quadratic Formula
For ax^2 + bx + c = 0, the solutions are x = (-b ± sqrt(b^2 – 4ac)) / 2a. Memorize this. It works on every quadratic, factorable or not.
The Discriminant
The expression b^2 – 4ac inside the quadratic formula is called the discriminant. It tells you:
- If positive: two real solutions.
- If zero: one real solution (vertex touches x-axis).
- If negative: no real solutions (parabola does not cross x-axis).
Completing the Square
Convert standard form to vertex form by completing the square. Useful for finding the vertex when factoring is hard.
Solving with Desmos
Desmos turns most quadratic questions into 5-second problems:
- Type the equation. Click the roots (x-intercepts). Click the vertex. Click the y-intercept.
- For word problems, convert to standard form, graph, and read the point you need.
- For discriminant questions, factor and check visually: does the parabola cross, touch, or miss the x-axis?
A Worked Example
Question: The function f(x) = x^2 – 6x + 5 has a minimum value at which value of x?
Algebra: Vertex x = -b / 2a = 6 / 2 = 3. Minimum at x = 3.
Desmos: Type y = x^2 – 6x + 5. Click the vertex. Read x = 3.
Common Quadratic Mistakes
- Forgetting that the parabola opens down when a is negative (affects max vs min).
- Sign errors when applying the quadratic formula.
- Confusing vertex form h with its opposite sign (vertex is at x = h, not -h).
- Forgetting to check if the equation is already in factored form.
Frequently Asked Questions
How many quadratic questions are on the SAT?
Typically 8 to 12 per math section. Quadratics appear in Advanced Math questions and sometimes in word problems.
Do I need to memorize the quadratic formula?
Yes. It is not on the reference sheet. Memorize it and the discriminant.
Can Desmos replace algebraic quadratic skills?
For most questions, yes. But knowing the algebra helps you recognize what Desmos is showing you and catch when a question is asking for an algebraic form.
What if the quadratic does not factor?
Use the quadratic formula or graph it in Desmos. Not all SAT quadratics factor cleanly.
Quadratics Are Rewarding to Master
Quadratic equations are one of the richest, most predictable SAT math topics. Master factoring, the quadratic formula, and vertex form. Learn to switch between the three forms quickly. Use Desmos to double-check algebra. Students who commit one focused week to quadratics often see their Advanced Math subscore jump noticeably.
Want deep quadratics coaching? Ayşenur specializes in Advanced Math for SAT. Request a free info session or WhatsApp +90 544 915 91 00.
Author: Ayşenur Özkan, Mathematics Instructor and SAT Math Tutor.
