Algebra Formula Sheet

A quick-reference sheet of the most important algebra formulas. Covers linear equations, quadratics, exponents, logarithms, factoring, and more. Each formula links to its full definition page with worked examples.

Linear Equations

Slope Formula

m = \frac{y_2 - y_1}{x_2 - x_1}

Slope-Intercept Form

y = mx + b

Point-Slope Form

y - y_1 = m(x - x_1)

Standard Form

Ax + By = C

Midpoint Formula

M = \left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\right)

Distance Formula

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Quadratics

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant

\Delta = b^2 - 4ac

Vertex Form

y = a(x-h)^2 + k

Vertex of a Parabola

h = -\frac{b}{2a}

Sum of Roots

x_1 + x_2 = -\frac{b}{a}

Product of Roots

x_1 \cdot x_2 = \frac{c}{a}

Exponent Rules

Product Rule

a^m \cdot a^n = a^{m+n}

Quotient Rule

\frac{a^m}{a^n} = a^{m-n}

Power Rule

(a^m)^n = a^{mn}

Zero Exponent

a^0 = 1 \quad (a \neq 0)

Negative Exponent

a^{-n} = \frac{1}{a^n}

Fractional Exponent

a^{m/n} = \sqrt[n]{a^m}

Factoring Identities

Difference of Squares

a^2 - b^2 = (a+b)(a-b)

Perfect Square Trinomial

a^2 \pm 2ab + b^2 = (a \pm b)^2

Sum of Cubes

a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Difference of Cubes

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Logarithm Properties

Definition

\log_b(x) = y \iff b^y = x

Product Rule

\log_b(MN) = \log_b M + \log_b N

Quotient Rule

\log_b\!\left(\frac{M}{N}\right) = \log_b M - \log_b N

Power Rule

\log_b(M^p) = p\,\log_b M

Change of Base

\log_b x = \frac{\ln x}{\ln b}

Sequences & Series

Arithmetic Sequence

a_n = a_1 + (n-1)d

Arithmetic Series

S_n = \frac{n}{2}(a_1 + a_n)

Geometric Sequence

a_n = a_1 \cdot r^{n-1}

Geometric Series (finite)

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

Infinite Geometric Series

S = \frac{a_1}{1 - r},\;|r|<1