Quadratics and Polynomials in Context

The SAT's Approach to Advanced Math

Advanced Math is tied with Algebra for the largest SAT Math domain at 35%. The core skill: manipulating nonlinear expressions and equations that model real situations. You will never see a bare "factor this polynomial" — every question has a scenario or asks you to interpret a feature of the equation.

Quadratic Equations — Three Solution Methods

Every SAT student needs all three methods because the test designs questions to reward the right tool.

Method	When to Use	Speed
Factoring	When the quadratic factors neatly over integers	Fastest
Completing the square	When the question asks for vertex form or min/max	Medium
Quadratic formula	When factoring is messy or the discriminant matters	Reliable fallback

Factoring

To factor $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.

Example: $x^2 - 5x + 6 = 0$

Find two numbers that multiply to $6$ and add to $-5$: those are $-2$ and $-3$.

$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$

Solutions: $x = 2$ or $x = 3$.

Completing the Square

Rewrite $ax^2 + bx + c$ into $a(x - h)^2 + k$ to find the vertex $(h, k)$.

Example: $x^2 + 8x + 10$

Take half of $8$, square it: $(8/2)^2 = 16$.

$x^2 + 8x + 16 - 16 + 10 = (x + 4)^2 - 6$

Vertex: $(-4, -6)$. The minimum value of the expression is $-6$.

The Discriminant

The discriminant $b^2 - 4ac$ tells you how many real solutions exist:

Discriminant	Solutions	Geometric Meaning
$b^2 - 4ac > 0$	Two distinct real solutions	Parabola crosses x-axis twice
$b^2 - 4ac = 0$	One repeated real solution	Parabola touches x-axis once
$b^2 - 4ac < 0$	No real solutions	Parabola doesn't cross x-axis

SAT Pattern: "For what value of $k$ does the equation $x^2 + kx + 9 = 0$ have exactly one solution?" Set discriminant to zero: $k^2 - 36 = 0$, so $k = \pm 6$.

Polynomial Zeros from Factored Form

If a polynomial is given as $f(x) = a(x - r_1)(x - r_2)(x - r_3)$, the zeros are $r_1, r_2, r_3$. The SAT tests this directly: "Which of the following is a zero of $f(x) = 2(x + 3)(x - 7)$?" Answer: $x = -3$ or $x = 7$.

SAT Trap: The factors show $(x + 3)$, so the zero is $-3$, not $+3$. At least one distractor will be the sign-flipped version.