SAT Quadratics: Every Question Type and How to Solve It
SAT quadratics show up in a handful of predictable forms. Learn each question type, the fastest solving method, and worked examples for the digital SAT.
SAT quadratics questions come in about eight predictable types: solving by factoring, using the quadratic formula, reading the vertex, using the discriminant, using sum and product of roots, matching equivalent forms, solving a line-parabola system, and modeling with a word problem. If you can recognize which type you are looking at, almost every quadratic on the digital SAT takes under a minute. Below is each type with a worked example and the fastest method.
Quadratics live in the Advanced Math domain, which also covers exponentials, polynomials, and nonlinear functions. For the full map of that domain, see our guide to SAT Advanced Math.
What forms of a quadratic does the SAT use?
Every quadratic question starts by handing you one of three forms. Each form makes one feature of the parabola obvious, and many questions are really asking, "Which form shows this feature?"
| Form | Looks like | What it shows instantly |
|---|---|---|
| Standard | y = ax² + bx + c | y-intercept is c; a tells you up (a > 0) or down (a < 0) |
| Factored | y = a(x − r)(x − s) | x-intercepts (zeros) are r and s |
| Vertex | y = a(x − h)² + k | vertex is (h, k); minimum or maximum value is k |
When a question asks which equivalent form "displays the minimum value as a constant or coefficient," the answer is vertex form. When it asks for the x-intercepts, it wants factored form.
Type 1: Solving a quadratic by factoring
If the numbers are small and a = 1, factor. Look for two numbers that multiply to c and add to b.
Example: Solve x² − 5x + 6 = 0.
- Find two numbers that multiply to 6 and add to −5: −2 and −3.
- Write (x − 2)(x − 3) = 0.
- Set each factor to zero: x = 2 or x = 3.
Always move everything to one side first. A common trap is an equation like x² = 4x, where students divide by x and lose the solution x = 0. Instead, write x² − 4x = 0, factor to x(x − 4) = 0, and keep both answers.
Type 2: When to use the quadratic formula
When factoring is not obvious within ten seconds, use the formula:
x = (−b ± √(b² − 4ac)) / (2a)
Example: Solve 2x² + 3x − 1 = 0.
- a = 2, b = 3, c = −1.
- b² − 4ac = 9 − 4(2)(−1) = 9 + 8 = 17.
- x = (−3 ± √17) / 4.
Answer choices with radicals are a strong signal that the formula, not factoring, is expected.
Type 3: Discriminant questions (how many solutions?)
The discriminant is the part under the square root: b² − 4ac. It tells you how many real solutions exist without solving.
| Discriminant | Number of real solutions | Graph |
|---|---|---|
| Positive | Two | Crosses the x-axis twice |
| Zero | Exactly one | Touches the x-axis at the vertex |
| Negative | None | Never touches the x-axis |
Example: For what value of c does x² + 6x + c = 0 have exactly one real solution?
Set the discriminant to zero: 6² − 4(1)(c) = 0, so 36 − 4c = 0 and c = 9.
The same idea answers questions about a line and a parabola intersecting at exactly one point. Substitute, rearrange into a single quadratic, and set the discriminant to zero.
Type 4: Finding the vertex, minimum, or maximum
The x-coordinate of the vertex is x = −b / (2a). Plug that back in to get the y-coordinate.
Example: What is the minimum value of y = x² − 6x + 5?
- x = −(−6) / (2·1) = 3.
- y = 9 − 18 + 5 = −4.
- The minimum value is −4, at the point (3, −4).
In vertex form this is y = (x − 3)² − 4. Expand it to check: x² − 6x + 9 − 4 = x² − 6x + 5. If a question asks you to rewrite in vertex form, completing the square or the −b/(2a) shortcut both work.
The vertex is also the midpoint of the two x-intercepts. If the zeros are 1 and 5, the vertex sits at x = 3. That symmetry solves many questions without any algebra.
Type 5: Sum and product of the solutions
For ax² + bx + c = 0:
- Sum of the solutions = −b / a
- Product of the solutions = c / a
Example: What is the sum of the solutions to 2x² − 8x + 3 = 0?
Sum = −(−8) / 2 = 4. You never need to find the individual solutions, which are messy here. This shortcut is one of the highest-value facts in SAT quadratics because it turns a two-minute problem into a ten-second one.
Type 6: Equivalent forms and expanding
Some questions show an expression like (3x − 2)² and ask which choice is equivalent. Expand carefully:
(3x − 2)² = 9x² − 12x + 4
The classic error is writing 9x² + 4 and forgetting the middle term. Remember (a − b)² = a² − 2ab + b².
Others ask you to factor a difference of squares: x² − 49 = (x − 7)(x + 7). Spot this whenever you see a perfect square minus a perfect square.
Type 7: Line and parabola systems
Example: The system y = x² and y = 2x + 3 intersects at which points?
- Set them equal: x² = 2x + 3.
- Rearrange: x² − 2x − 3 = 0.
- Factor: (x − 3)(x + 1) = 0, so x = 3 or x = −1.
- Find y: (3, 9) and (−1, 1).
If a question only asks how many intersection points exist, skip straight to the discriminant from Type 3. For a refresher on linear systems, see SAT systems of equations.
Type 8: Quadratic word problems
The most common setup is a projectile: height as a function of time.
Example: A ball's height in feet is h(t) = −16t² + 64t + 5, where t is seconds after launch. What is the maximum height?
- Time of maximum: t = −64 / (2·−16) = 2 seconds.
- Height: h(2) = −64 + 128 + 5 = 69 feet.
Know what each piece means in context. The constant 5 is the starting height (when t = 0). The positive zero is when the ball hits the ground. The vertex is the peak. SAT questions often ask "what does 5 represent?" rather than asking you to calculate anything.
How to use Desmos on SAT quadratics
The digital SAT includes a built-in Desmos graphing calculator, and it is excellent for quadratics. Type the equation, then click the points where the graph crosses the x-axis or reaches its vertex. Desmos displays the coordinates directly.
Desmos is fastest for:
- Finding vertices and intercepts of ugly quadratics
- Counting intersections of a line and a parabola
- Checking which answer choice is equivalent (graph two expressions and see whether they overlap)
It is slower than mental math for clean factoring or sum-of-roots questions. Use our walkthrough on how to use Desmos on the SAT to build the habit before test day.
Common SAT quadratics mistakes
- Dropping a solution. Dividing both sides by x removes x = 0.
- Sign errors in −b/(2a). If b is already negative, −b is positive.
- Forgetting the middle term when squaring a binomial.
- Answering the wrong question. The problem asks for the maximum height and you report the time. Reread the last line before choosing.
- Ignoring context. A negative time in a projectile problem gets thrown out.
How Ivystar can help
Quadratics are one of the most reliable places to pick up SAT points, because the question types repeat. In Ivystar SAT Prep, a tutor diagnoses exactly which of these types costs you time and drills them with real digital-SAT-style practice; our students improve by an average of 170 points. Try a free practice SAT or book a free 20-minute strategy call to see where you stand.
Frequently asked questions
How many quadratics questions are on the SAT?+
Quadratics fall under the Advanced Math domain, which makes up roughly 35 percent of the Math section. The exact number of quadratic questions varies by test, but expect several across the two Math modules.
Is the quadratic formula on the SAT reference sheet?+
No. The reference sheet covers geometry formulas such as area, volume, and special right triangles. You need to memorize the quadratic formula and the vertex shortcut x = -b/(2a).
Should I solve SAT quadratics by hand or with Desmos?+
Use mental math for clean factoring and sum-or-product questions, and Desmos for messy equations, vertices, and intersection counts. The best scorers switch between the two depending on the question.
What is the fastest way to find the sum of the solutions?+
Use -b/a from standard form ax^2 + bx + c = 0. You do not need to solve for each root individually.
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