Algebra has a reputation for being a long list of formulas to memorize. The truth is simpler — most of the algebra you will ever do relies on a small core of rules and identities that show up over and over again. This cheat sheet collects them in one place, organized by topic, so you can use it as a study reference, a homework guide, or a last-minute review before a test. Every rule includes a quick example so you can see it in action.
These are the foundational properties that govern how numbers and variables combine. They apply to all real numbers.
| Property | Rule | Example |
|---|---|---|
| Commutative (addition) | a + b = b + a | 3 + 7 = 7 + 3 |
| Commutative (multiplication) | a × b = b × a | 4 × 5 = 5 × 4 |
| Associative (addition) | (a + b) + c = a + (b + c) | (2 + 3) + 4 = 2 + (3 + 4) |
| Associative (multiplication) | (a × b) × c = a × (b × c) | (2 × 3) × 4 = 2 × (3 × 4) |
| Distributive | a(b + c) = ab + ac | 3(x + 5) = 3x + 15 |
| Identity (addition) | a + 0 = a | 7 + 0 = 7 |
| Identity (multiplication) | a × 1 = a | 9 × 1 = 9 |
| Inverse (addition) | a + (−a) = 0 | 5 + (−5) = 0 |
| Inverse (multiplication) | a × (1/a) = 1 (a ≠ 0) | 4 × (1/4) = 1 |
When evaluating any expression, follow this exact order. Many algebra mistakes come from rushing through this step.
Multiplication and division are same priority, evaluated left to right. So 12 ÷ 2 × 3 = (12 ÷ 2) × 3 = 18, not 12 ÷ 6 = 2.
| Rule | Formula | Example |
|---|---|---|
| Product | am × an = am+n | 2³ × 2² = 2⁵ = 32 |
| Quotient | am ÷ an = am−n | 5⁵ ÷ 5² = 5³ = 125 |
| Power | (am)n = am×n | (3²)³ = 3⁶ = 729 |
| Product of powers | (ab)n = anbn | (2x)³ = 8x³ |
| Quotient of powers | (a/b)n = an/bn | (3/4)² = 9/16 |
| Zero exponent | a0 = 1 (a ≠ 0) | 170 = 1 |
| Negative exponent | a−n = 1/an | 2−3 = 1/8 |
| Fractional exponent | a1/n = n√a | 81/3 = 2 |
| General fractional | am/n = n√(am) | 82/3 = 4 |
For deeper practice with exponent rules, see our exponents practice page.
For any equation of the form ax + b = c:
| Form | Equation | What it shows |
|---|---|---|
| Slope-Intercept | y = mx + b | slope m, y-intercept b |
| Point-Slope | y − y₁ = m(x − x₁) | slope m through point (x₁, y₁) |
| Standard | Ax + By = C | x-intercept C/A, y-intercept C/B |
| Horizontal line | y = c | slope 0 |
| Vertical line | x = c | slope undefined |
Recognizing these patterns saves enormous time on tests.
| Pattern | Factored Form | Example |
|---|---|---|
| Difference of squares | a² − b² = (a + b)(a − b) | x² − 9 = (x + 3)(x − 3) |
| Perfect square (positive) | a² + 2ab + b² = (a + b)² | x² + 6x + 9 = (x + 3)² |
| Perfect square (negative) | a² − 2ab + b² = (a − b)² | x² − 8x + 16 = (x − 4)² |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | x³ + 8 = (x + 2)(x² − 2x + 4) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | x³ − 27 = (x − 3)(x² + 3x + 9) |
| Standard trinomial | x² + (a+b)x + ab = (x + a)(x + b) | x² + 7x + 12 = (x + 3)(x + 4) |
For any quadratic equation ax² + bx + c = 0 (where a ≠ 0):
The discriminant (b² − 4ac) tells you the type of solutions:
For deeper practice, see our quadratic equations page.
A parabola written as y = a(x − h)² + k has:
Converting from standard form ax² + bx + c, the vertex x-coordinate is at x = −b/(2a).
Three methods to solve two equations in two variables:
A system has one solution if the lines cross, no solution if parallel, infinitely many solutions if identical. See our systems practice.
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. This is the single most common cause of inequality mistakes.
Otherwise, solve inequalities exactly like equations. Practice at our inequalities page.
If f(x) = 3x + 2, then:
| What | Formula |
|---|---|
| Distance formula | d = √((x₂ − x₁)² + (y₂ − y₁)²) |
| Midpoint formula | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) |
| Pythagorean theorem | a² + b² = c² |
| Sum of arithmetic series | S = n(a₁ + an)/2 |
| Compound interest | A = P(1 + r/n)nt |
Print this page or bookmark it. The most effective study method is to glance at it for thirty seconds before each homework or practice session — not to memorize the whole thing at once, but to refresh the concepts you are about to use. After two or three weeks of daily reference, most of these rules will be in long-term memory and you will need the cheat sheet less and less. That is the goal: use it now so you can stop needing it later.
Eventually yes — but not all at once. Start with PEMDAS, the basic exponent rules, and the slope formula. Add the quadratic formula and the difference-of-squares pattern next. The rest comes from regular use. Memorization through repeated application beats flashcards every time.
Most teachers do not allow them, but standardized tests (SAT, ACT) provide a small built-in formula sheet at the start of the math section. Knowing whether your test allows a reference sheet determines how much you need to memorize. Even when allowed, knowing the formulas by heart is faster than flipping through.
For high-school algebra, the quadratic formula is the single most useful single equation. After that, the slope formula and the distance formula are tied for second. The exponent rules collectively are the most useful set — they show up in almost every algebra topic.
Trigonometry is typically covered in a separate course (sometimes called precalculus or Algebra 2/Trig). This sheet covers what is technically considered algebra. For an introduction to trig topics, see our pre-calculus page.
Yes — this cheat sheet is free to print for personal study use. Use your browser print function (File → Print or Ctrl/Cmd + P) to get a clean printable version. Tables typically print well; you may want to deselect background colors in your print dialog.
Each section above links to a dedicated practice page on the topic. For example, quadratic equations, inequalities, and exponents each have full lessons with worked examples and practice problems.
Apply these formulas with interactive practice.
Start Algebra Practice →