A one-page algebra reference: order of operations, exponent and radical rules, factoring patterns, the quadratic formula, line equations, and logarithm rules.
Order of operations (PEMDAS)
Work in this order: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). See the Order of Operations Guide.
Exponent rules
| Rule | Formula |
|---|---|
| Product | xa · xb = xa+b |
| Quotient | xa / xb = xa−b |
| Power of a power | (xa)b = xab |
| Power of a product | (xy)a = xaya |
| Zero exponent | x0 = 1 |
| Negative exponent | x−a = 1 / xa |
| Fractional exponent | x1/n = n√x |
Radical rules
| Rule | Formula |
|---|---|
| Product of roots | √(ab) = √a · √b |
| Quotient of roots | √(a/b) = √a / √b |
| Root squared | (√a)² = a |
| Rationalize | 1/√a = √a / a |
Factoring patterns
| Pattern | Factored form |
|---|---|
| Difference of squares | a² − b² = (a + b)(a − b) |
| Perfect square (sum) | a² + 2ab + b² = (a + b)² |
| Perfect square (difference) | a² − 2ab + b² = (a − b)² |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
Quadratics
| Topic | Formula |
|---|---|
| Standard form | ax² + bx + c = 0 |
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a |
| Discriminant | b² − 4ac |
| Discriminant test | > 0: two roots = 0: one < 0: none real |
Lines and linear equations
| Form | Equation |
|---|---|
| Slope-intercept | y = mx + b |
| Point-slope | y − y₁ = m(x − x₁) |
| Standard | Ax + By = C |
| Slope | m = (y₂ − y₁) / (x₂ − x₁) |
Logarithm rules
| Rule | Formula |
|---|---|
| Product | log(xy) = log x + log y |
| Quotient | log(x/y) = log x − log y |
| Power | log(xn) = n log x |
| Identity | logb(b) = 1, logb(1) = 0 |
| Change of base | logb(x) = log x / log b |
Related guides: Exponents, Factoring, Logarithms, and the Quadratic Formula.