Quadratics & Polynomials

1. Introduction

Quadratics are the heart of the SAT's Passport to Advanced Math domain, and they reward a deep, flexible understanding more than any other topic on the test. A quadratic is any equation whose highest power is $x^2$, and its graph is a parabola — a symmetric U-shaped curve. The College Board loves quadratics because a single curve can be described three different ways, and each description hands you different information for free. The whole game is learning to choose the right form for the question being asked.

Students who memorize only the quadratic formula leave points on the table. The strongest test-takers can look at a parabola problem and instantly decide: "This asks for the vertex, so I want vertex form" or "This asks where it crosses the $x$-axis, so I want factored form." They also know when not to compute at all — for instance, reading the number of real solutions straight off the discriminant.

This article develops every quadratic skill the SAT tests: the three forms and what each reveals, factoring, the quadratic formula, completing the square, the discriminant, the vertex and axis of symmetry, and the bridge from quadratics to higher-degree polynomials through the factor–root connection. We finish with SAT-specific strategies for the trap questions that hinge on signs and on counting solutions.

2. Core Concepts

2.1 What makes an equation quadratic

A quadratic has the form $f(x) = ax^2 + bx + c$ with $a \neq 0$. The $a \neq 0$ condition matters: if $a = 0$ the $x^2$ term vanishes and the equation is merely linear. The sign of $a$ controls the parabola's direction — $a > 0$ opens upward (has a minimum) and $a < 0$ opens downward (has a maximum). The magnitude of $a$ controls how narrow or wide the curve is.

2.2 The three equivalent forms

Every quadratic can be written three ways, all describing the same parabola:

• Standard form $ax^2 + bx + c$ — the constant $c$ is the $y$-intercept (set $x = 0$). Best for reading where the curve hits the $y$-axis and for applying the quadratic formula.
• Factored form $a(x - r_1)(x - r_2)$ — $r_1$ and $r_2$ are the zeros / $x$-intercepts. Best for finding roots and sketching where the graph crosses the $x$-axis.
• Vertex form $a(x - h)^2 + k$ — $(h, k)$ is the vertex, the highest or lowest point. Best for maxima/minima and for graphing the turning point.

2.3 The vertex and axis of symmetry

A parabola is perfectly symmetric about a vertical line through its vertex, the axis of symmetry. Its equation is

$x = -\frac{b}{2a}$

which is also the $x$-coordinate of the vertex. To get the vertex's $y$-coordinate, substitute that $x$ back into the function. Because of symmetry, the two roots (when real) are equidistant from this line — a fact that lets you find a missing root quickly.

2.4 The discriminant: counting solutions without solving

The expression under the radical in the quadratic formula, $\Delta = b^2 - 4ac$, is the discriminant. It reveals the number of real solutions before you do any solving:

• $\Delta > 0$: two distinct real solutions (parabola crosses the $x$-axis twice).
• $\Delta = 0$: exactly one real solution, a repeated root (the vertex sits on the $x$-axis).
• $\Delta < 0$: no real solutions (the parabola never touches the $x$-axis).

SAT questions frequently ask "for what value of $c$ does this have exactly one solution?" — that is a discriminant-equals-zero problem in disguise.

2.5 Factors, roots, and polynomials

The single most important idea linking quadratics to higher-degree polynomials is the factor–root connection: $(x - r)$ is a factor of a polynomial $f(x)$ exactly when $r$ is a root (i.e. $f(r) = 0$). The Remainder Theorem generalizes this: dividing $f(x)$ by $(x - r)$ leaves a remainder of $f(r)$. So a remainder of $0$ confirms $r$ is a root. This connection lets you move freely between a polynomial's graph (its $x$-intercepts), its factored form, and its values.

2.6 The zero-product property

If a product of factors equals zero, at least one factor must be zero: $(x - 2)(x + 5) = 0 \Rightarrow x = 2$ or $x = -5$. This is the basis of solving by factoring. Critical SAT rule: the equation must be set equal to zero before you apply this — move everything to one side first.

2.7 Vieta's formulas (sum and product of roots)

For $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:

$r_1 + r_2 = -\frac{b}{a}, \qquad r_1 r_2 = \frac{c}{a}$

The SAT asks for these sums and products without asking for the roots themselves. If you see "sum of solutions," go to Vieta before the quadratic formula.

2.8 Completing the square as a vertex tool

Completing the square rewrites $ax^2 + bx + c$ as $a(x - h)^2 + k$. On the SAT, you rarely need the full ritual — but knowing that $x^2 + 6x + \underline{9}$ completes to $(x + 3)^2$ helps you read vertex form and convert standard to vertex when asked.

2.9 Quadratic vs. quadratic function vs. quadratic equation

• Function $f(x) = x^2 - 4x + 3$ — you can evaluate, graph, find vertex.
• Equation $x^2 - 4x + 3 = 0$ — you solve for $x$ (the zeros). The SAT blurs these: "What is the minimum of $y = x^2 - 4x + 3$?" is a function question; "For what $x$ is $y = 0$?" is an equation question on the same parabola.

2.10 Higher-degree polynomials on the SAT

Cubic and quartic polynomials appear in factored form: $f(x) = (x - 1)(x + 2)(x - 3)$. Zeros are $1$, $-2$, $3$. The SAT may ask for $f(0)$ (multiply constants) or whether $x = 1$ is a zero (yes — factor $(x - 1)$). Degree-3 graphs can have up to three $x$-intercepts.

2.11 Projectile and area models

Classic SAT contexts: height $h(t) = -16t^2 + v_0 t + h_0$ (feet, seconds) and area as a quadratic in one dimension. The negative leading coefficient means a maximum — the peak height at the vertex. Time to hit the ground: solve $h(t) = 0$ for $t > 0$.

2.12 Systems involving quadratics

A line and a parabola intersect when $mx + b = ax^2 + cx + d$. Rearrange to a quadratic and solve — $0$, $1$, or $2$ intersection points match the discriminant of that quadratic. The SAT may only ask for the count, not the coordinates.

2.13 The graph of $y = a(x - r)^2$ vs. $y = ax^2 + bx + c$

Both describe parabolas, but vertex form reveals the turning point immediately. Standard form reveals the $y$-intercept. Converting between them is completing the square — a skill worth practicing until automatic.