Quadratics anchor the Advanced Math domain, and nearly every hard SAT Math question that isn't a linear system is a quadratic in disguise. The topic looks large, but it collapses to a single idea: a quadratic hands you information for free depending on which of three forms it's written in. Learn to read the form and you've learned most of the topic.
The three forms — and what each reveals
A quadratic is any equation whose highest power is x². Its graph is a parabola. The same parabola can be written three ways, and each way exposes a different feature at a glance:
| Form | Looks like | Reveals for free |
|---|---|---|
| Standard | ax² + bx + c | y-intercept is c; feeds the quadratic formula |
| Factored | a(x − r)(x − s) | x-intercepts (roots) are r and s |
| Vertex | a(x − h)² + k | vertex is (h, k) — the maximum or minimum |
The single most useful test-day skill is matching the question to the form: asked for the minimum value? Convert to vertex form. Asked where the graph crosses the x-axis? Factor. Asked for the y-intercept? Read c from standard form.
Factoring and the roots
Factoring rewrites ax² + bx + c as a product that equals zero, and a product is zero only when one of its parts is zero. For a simple monic quadratic (a = 1), find two numbers that multiply to c and add to b.
Worked example — factoring
Solve x² − 7x + 12 = 0.
Two numbers multiply to 12 and add to −7 → −3 and −4. So it factors as (x − 3)(x − 4) = 0, giving x = 3 or x = 4.
Fast check: the sum of the roots is −b/a = 7 and the product is c/a = 12. 3 + 4 = 7 and 3 × 4 = 12 ✓.
That sum-and-product relationship (roots sum to −b/a, multiply to c/a) also answers "what is the sum of the solutions" questions without solving.
The vertex: maximum, minimum, and symmetry
The vertex is the turning point of the parabola. From standard form, its x-coordinate is x = −b / (2a); plug that back in for the y-coordinate. The parabola opens up when a > 0 (vertex is a minimum) and down when a < 0 (vertex is a maximum).
- The axis of symmetry is the vertical line x = −b/(2a). The two roots are always mirror images across it — so the axis is exactly halfway between them.
- If you know both roots, the vertex's x-coordinate is simply their average.
- On the Digital SAT you can graph y = ax² + bx + c in Desmos and click the vertex dot to read (h, k) exactly.
The discriminant: how many real solutions?
The discriminant is b² − 4ac, the expression under the square root in the quadratic formula. Its sign alone tells you how many real solutions exist — no solving required:
| b² − 4ac | Real solutions | Graph vs. x-axis |
|---|---|---|
| Positive | Two distinct | Crosses at two points |
| Zero | Exactly one (repeated) | Just touches (tangent) |
| Negative | None (two complex) | Never touches |
This is why so many SAT questions ask for the coefficient that makes an equation have exactly one real solution — that's just b² − 4ac = 0 solved for the unknown.
The quadratic formula (not on the reference sheet)
When a quadratic won't factor cleanly, use x = (−b ± √(b² − 4ac)) / (2a). It is not printed on the Bluebook reference sheet, so memorize it — but on the Digital SAT you can often skip it entirely by graphing and clicking the x-intercepts. Keep the formula as a backup for symbolic questions whose answers are written in terms of letters, where graphing doesn't help.
For Chinese-American families · 华人家庭视角
Students who struggle with quadratics almost always have a form-recognition gap, not an algebra gap: they can factor when told to, but don't see that "minimum value" means "vertex form" or that "one solution" means "discriminant = 0." Our diagnostic pinpoints exactly which of these translations is missing.
卡在二次函数的学生，几乎都是识别形式的能力缺口，而不是代数计算 缺口：让他因式分解他会，但看不出"最小值"="顶点式"、 "只有一个解"="判别式=0"。我们的诊断会精确定位到底是哪一步 翻译没打通。
Quick reference
- Standard ax² + bx + c → y-intercept c; factored a(x − r)(x − s) → roots r, s; vertex a(x − h)² + k → vertex (h, k).
- Vertex x-coordinate = −b/(2a); axis of symmetry is x = −b/(2a); roots are symmetric about it.
- Sum of roots = −b/a, product = c/a.
- Discriminant b² − 4ac: positive → 2 real, zero → 1 real, negative → none. 'Exactly one solution' means b² − 4ac = 0.
- Quadratic formula x = (−b ± √(b² − 4ac)) / (2a) — memorize it; it's not on the reference sheet.
Frequently asked questions
What is the discriminant and what does it tell me on the SAT?
The discriminant is b² − 4ac, the part under the square root in the quadratic formula. Its sign tells you the number of real solutions without solving: positive means two distinct real solutions, zero means exactly one (a repeated root, where the parabola just touches the x-axis), and negative means no real solutions (two complex ones). SAT questions often ask for the value of a coefficient that makes an equation have 'exactly one real solution' — that is the same as setting b² − 4ac = 0.
Which quadratic form should I use — standard, factored, or vertex?
Use whichever the answer needs. Standard form ax² + bx + c reveals the y-intercept (c) and feeds the quadratic formula. Factored form a(x − r)(x − s) reveals the x-intercepts (roots) r and s instantly. Vertex form a(x − h)² + k reveals the vertex (h, k), which is the maximum or minimum. If a question asks for the minimum value, convert to vertex form; if it asks for where the graph crosses the x-axis, factor.
How do I find the vertex from standard form quickly?
The x-coordinate of the vertex is x = −b / (2a). Plug that back into the equation to get the y-coordinate. The axis of symmetry is the vertical line x = −b/(2a), and the two roots are always symmetric about it — so if you know one root and the axis, you know the other. On the Digital SAT you can also just graph y = ax² + bx + c in Desmos and click the vertex point.
Do I need to memorize the quadratic formula for the SAT?
It helps, because it is not on the Bluebook reference sheet. The formula is x = (−b ± √(b² − 4ac)) / (2a). That said, on the Digital SAT you can graph the equation in Desmos and click the x-intercepts to read the solutions, which avoids arithmetic slips. Memorize the formula as a backup for symbolic questions (answers written in terms of letters) where graphing does not help.
What is the sum and product of the roots trick?
For ax² + bx + c = 0, the two roots add up to −b/a and multiply to c/a. This lets you check factoring fast or answer 'what is the sum of the solutions' without solving. Example: for x² − 7x + 12 = 0, the roots sum to 7 and multiply to 12, which points straight at 3 and 4.
How Lighthouse Math teaches this
Lighthouse Math (灯塔数学) is a diagnostic-driven SAT Math program for Chinese-American families in the Boston / Arlington MA area. We don't assign more volume — we map the two or three specific structural mechanisms silently capping a student's score, then close them. Every concept above is taught bilingually so the underlying math is understood, not just translated.
我们用中英双语讲清机制本身，而不是靠刷题量。先做诊断，找到隐形封顶你孩子分数的 2–3 个结构性卡点，再逐一打破 —— 让孩子真正理解数学，而不只是背题型。