SAT Math Advanced Math: Algebra Beyond Lines

Equivalent expression

An algebraic expression written in a different form but equal in value for every allowed input.

Domain

The set of input values for which an expression or function is defined.

Distributive property

The rule that multiplies a term across a sum or difference, such as $a(b + c) = ab + ac$.

Exponent rules

Laws for working with powers, such as $a^m \cdot a^n = a^{m+n}$, used only when any required restrictions are satisfied.

Difference of squares

A factoring pattern: $a^2 - b^2 = (a - b)(a + b)$.

Perfect square trinomial

A trinomial that factors into a squared binomial, such as $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.

Rational expression

A fraction whose numerator and denominator are polynomials.

Completing the square

A method of rewriting a quadratic to create a perfect square trinomial, often to find vertex form.

Nonlinear equation

An equation whose graph is not a straight line, such as a quadratic, rational, or radical equation.

Quadratic equation

An equation of the form $ax^2 + bx + c = 0$, where a is not 0.

Quadratic formula

A formula for solving any quadratic equation: $x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)$.

Discriminant

The expression $b^2 - 4ac$, which tells how many real solutions a quadratic equation has.

Rational equation

An equation that contains rational expressions and must be solved while keeping denominator restrictions in mind.

Radical equation

An equation that contains a radical and is often solved by isolating the radical, squaring both sides, and checking the result.

Extraneous solution

A value produced during solving that does not actually satisfy the original equation.

Nonlinear system

A system of equations in which at least one equation is nonlinear.

Substitution

A method for solving a system by replacing one variable with an equivalent expression from another equation.

Nonlinear function

A function whose graph is not a straight line.

Quadratic function

A function of the form $f(x) = ax^2 + bx + c$ whose graph is a parabola.

Standard form

The quadratic form $f(x) = ax^2 + bx + c$, which makes the y-intercept easy to identify.

Factored form

The quadratic form $a(x - r_1)(x - r_2)$, which makes the zeros or x-intercepts easy to read.

Vertex form

The quadratic form $a(x - h)^2 + k$, which shows the vertex at (h, k).

Polynomial

An expression made of terms like $a_n x^n$ where the exponents are nonnegative integers.

Rational function

A function written as $p(x) / q(x)$; values that make $q(x) = 0$ are excluded from the domain and may create asymptotes or holes.

Radical function

A function that contains a radical, such as $\sqrt{x - 3}$, with a domain limited by the value inside the radical.