SAT Quadratics: Every Question Type

Every quadratic on the SAT is the same parabola wearing a different outfit. Standard form: ax² + bx + c. Factored form: a(x - p)(x - q). Vertex form: a(x - h)² + k. The question almost never tells you which one you're looking at. It just expects a different move depending on the shape in front of you, and reaching for the wrong move burns time that doesn't come back.

Here's the shortcut that saves the most of it: don't convert a quadratic into a form you don't need. Handed factored form and asked for the roots? Read them off, done. Handed vertex form and asked for a minimum? Same thing, no work required. Converting out of habit, before even checking what's being asked, is where a lot of the wasted seconds on this test actually go.

Standard form looks like x² - 2x - 15 = 0, and it's usually the one that takes the most work, since the roots aren't sitting in plain sight yet. Factoring means finding two numbers that multiply to -15 and add to -2. Here that's -5 and 3, so (x - 5)(x + 3) = 0 and x = 5 or x = -3. Check it by multiplying back out: (x - 5)(x + 3) = x² + 3x - 5x - 15 = x² - 2x - 15. Matches.

Not every standard-form quadratic breaks into clean whole numbers, and that's fine. When it doesn't, the quadratic formula always works: x = (-b ± √(b² - 4ac)) / (2a). It's slower than factoring, so try factoring first on anything that looks like it has whole-number roots. Switch to the formula only once a couple of tries at factoring turn up nothing.

Vertex form is a(x - h)² + k, and the payoff lands immediately: the vertex sits right at (h, k). Take f(x) = 2(x - 3)² + 1. No solving needed. The vertex is (3, 1), and since a = 2 is positive, that point is a minimum, not a maximum. Expand it to double-check: 2(x - 3)² + 1 = 2(x² - 6x + 9) + 1 = 2x² - 12x + 19, the exact same parabola in standard form, just dressed differently.

Getting from standard form to vertex form runs through completing the square. Take x² + 6x + 5. Take half of 6, square it, and that's 9: x² + 6x + 9 - 9 + 5, which regroups into (x + 3)² - 4. Vertex form puts the vertex at (-3, -4). Check it against the roots: x² + 6x + 5 factors to (x + 1)(x + 5), so the roots are -1 and -5, and a parabola's vertex always sits exactly halfway between its roots. Halfway between -1 and -5 is -3. Matches.

Sitting inside the quadratic formula is b² - 4ac, called the discriminant, and it answers a question before any solving starts: how many real solutions does this equation have? Positive means two real roots. Zero means exactly one, a repeated root. Negative means none among real numbers. Take x² - 4x + 7 = 0: the discriminant is (-4)² - 4(1)(7) = 16 - 28 = -12. Negative, so this one has no real solutions, and grinding through the rest of the formula would only confirm that the slow way.

Compare it to x² - 6x + 9 = 0, where the discriminant is (-6)² - 4(1)(9) = 36 - 36 = 0. Exactly one repeated root. Factor it and the reason is obvious: x² - 6x + 9 = (x - 3)², so x = 3 is the only solution, just counted twice. On a graph, that's a parabola grazing the x-axis at one point instead of crossing it. A positive discriminant crosses the x-axis twice, zero touches it once, negative never touches it at all.

A lot of quadratic questions on the SAT come dressed as a story: height, area, revenue. Say a ball launched straight up has height in feet after t seconds given by h(t) = -16t² + 64t. Factored form makes the landing question quick: h(t) = -16t(t - 4), so h = 0 at t = 0 and t = 4. It launches at t = 0 and lands at t = 4 seconds, which checks out since those are the only two moments its height is zero.

Finding the peak height turns out to be a vertex question in disguise, even though the word vertex never appears in the problem. The t-coordinate of the vertex is -b / (2a) = -64 / (2 × -16) = 2 seconds, and plugging that back in gives h(2) = -16(4) + 64(2) = -64 + 128 = 64 feet. A context question like this one is just standard form, factored form or the vertex formula in a costume, so the same moves from earlier still apply. If this kind keeps causing trouble, StudyPebble's practice by skill drills nonlinear equations and nonlinear functions on their own, with an explanation attached to every question, and Desmos graphs any of these instantly for a quick visual check.