SAT Math: Advanced Math

Equivalent expressions

These questions ask you to rewrite an expression in an equivalent form, often by factoring, expanding, or simplifying a rational expression. The SAT usually presents four answer choices that all look plausible, so the safest strategy is to fully simplify the given expression yourself before comparing it to the choices, rather than trying to manipulate each choice to match the original.

For rational expressions (fractions with variables), factor both the numerator and denominator first to see which factors cancel, and note any values of the variable that would make the original denominator zero, since those values must be excluded from the domain.

Solving quadratic equations

Quadratics can be solved by factoring, completing the square, or the quadratic formula. Factoring is fastest when the equation has integer roots: look for two numbers that multiply to the constant term and add to the coefficient of the linear term. When factoring isn't obvious, the quadratic formula always works and is worth having memorized.

The discriminant, b² − 4ac, tells you the number of real solutions without fully solving the equation: positive means two distinct real solutions, zero means exactly one real solution (a repeated root), and negative means no real solutions. The SAT often asks about the discriminant directly, for example asking for a value of a coefficient that makes an equation have exactly one solution.

Completing the square and vertex form

Completing the square rewrites a quadratic in vertex form, a(x − h)² + k, which directly reveals the vertex (h, k) of the parabola — useful for questions asking for a maximum or minimum value without needing calculus. To complete the square on x² + bx, add and subtract (b/2)² to create a perfect square trinomial.

Nonlinear systems of equations

A nonlinear system typically pairs a linear equation with a quadratic (or another curve). Substitution is usually the most reliable method: solve the linear equation for one variable, then substitute into the nonlinear equation to reduce it to a single-variable equation you can solve directly. Remember that such a system can have zero, one, or two solutions, depending on whether the line intersects the curve, is tangent to it, or misses it entirely.

Exponential and radical functions

Exponential functions of the form y = a·bˣ model growth or decay, where a is the starting value and b is the growth factor per unit of x (b > 1 for growth, 0 < b < 1 for decay). Radical equations involve a variable under a square or cube root; solve by isolating the radical and raising both sides to the appropriate power, then always check your solution in the original equation, since squaring both sides can introduce extraneous solutions that don't actually satisfy the original radical equation.

Function notation and transformations

Function notation questions ask you to evaluate f(a) for a given input, combine functions such as f(g(x)), or interpret transformations like f(x) + k (vertical shift), f(x − h) (horizontal shift), or −f(x) (reflection). Read function notation literally and mechanically: f(3) means substitute 3 everywhere the input variable appears in the function's definition, and f(x) + 2 shifts every output up by 2, while f(x + 2) shifts every input — and therefore the whole graph — left by 2.