"Algebra formulas" is typed by students the night before a test and by parents trying to help with homework, and the lists that come up are either twenty formulas with no context or two hundred with no order. This page gives every formula a student meets between Grade 8 (CBSE Class 8) and Algebra 2 (Class 11, A-level year 1), grouped by the job it does, with the situation it is for and the mistake most often made with it, because a formula is useless without the second and dangerous without the third. The printable sheet has all of them on one page, with the names used in India, the UK and the US.
Download the algebra formulas sheet (PDF) (every formula on this page grouped by job, with the names used in each system, and the twelve most common mistakes on the back; two pages, free).
1. The laws of exponents
| Formula | Use it when | The mistake |
|---|---|---|
| aᵐ × aⁿ = aᵐ⁺ⁿ | Multiplying powers of the same base | Multiplying the exponents |
| aᵐ ÷ aⁿ = aᵐ⁻ⁿ | Dividing powers of the same base | Dividing the exponents |
| (aᵐ)ⁿ = aᵐⁿ | A power of a power | Adding the exponents |
| (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ | A power of a product or quotient | Applying it to a sum: (a + b)ⁿ is not aⁿ + bⁿ |
| a⁰ = 1 (a ≠ 0) | Anything to the power zero | Writing 0 |
| a⁻ⁿ = 1/aⁿ | Negative exponents | Making the answer negative |
| a¹ᐟⁿ = ⁿ√a, aᵐᐟⁿ = (ⁿ√a)ᵐ | Rational exponents (Algebra 1 unit 6, Class 9) | Taking the root of the exponent |
2. The algebraic identities (the factoring patterns)
| Identity | Use it when | The mistake |
|---|---|---|
| (a + b)² = a² + 2ab + b² | Expanding or completing the square | Writing a² + b² (the missing middle term is the commonest error in all of algebra) |
| (a − b)² = a² − 2ab + b² | Same, with a minus | Getting −b² |
| (a + b)(a − b) = a² − b² | Factoring a difference of squares; rationalising | Trying to factor a² + b² (it does not factor over the reals) |
| (x + a)(x + b) = x² + (a + b)x + ab | Factoring trinomials with leading coefficient 1 | Sign errors: the two numbers must add to the middle and multiply to the last |
| (a + b)³ = a³ + 3a²b + 3ab² + b³ | CBSE Class 9; binomial expansions | Dropping the 3s |
| (a − b)³ = a³ − 3a²b + 3ab² − b³ | Same | The alternating signs |
| a³ + b³ = (a + b)(a² − ab + b²) | Factoring a sum of cubes | The middle sign of the second bracket is opposite to the first |
| a³ − b³ = (a − b)(a² + ab + b²) | Factoring a difference of cubes | Same |
| (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca | CBSE Class 9; three-term squares | Missing one of the three cross terms |
3. Quadratics
| Formula | Use it when | The mistake |
|---|---|---|
| Standard form ax² + bx + c = 0 | Before anything else: move every term to one side | Using the formula with c on the other side |
| x = [−b ± √(b² − 4ac)] / 2a | Any quadratic, always works | Dividing only the root by 2a; forgetting −b is positive when b is negative |
| Discriminant D = b² − 4ac | Deciding how many real roots: D > 0 two, D = 0 one, D < 0 none | Computing −4ac with the wrong sign |
| Sum of roots = −b/a, product = c/a | CBSE Class 10; checking answers; building a quadratic from its roots | Dropping the minus on the sum |
| Vertex x = −b/2a; vertex form y = a(x − h)² + k | Maximum and minimum problems; graphing | The sign of h: (x − 3)² has vertex at x = +3 |
| Completing the square: x² + bx = (x + b/2)² − (b/2)² | Deriving the formula; circle equations; vertex form | Forgetting to subtract (b/2)² |
The three solving methods, with worked examples, are in how to solve quadratic equations step by step.
4. Lines and the coordinate plane
| Formula | Use it when | The mistake |
|---|---|---|
| Slope m = (y₂ − y₁)/(x₂ − x₁) | Any two points | Subtracting in different orders on top and bottom |
| Slope-intercept y = mx + b (US) or y = mx + c (India, UK) | Graphing; reading the rate and the start | Reading b as the x-intercept |
| Point-slope y − y₁ = m(x − x₁) | A point and a slope | Sign of the point's coordinates |
| Standard form Ax + By = C | Systems; intercepts | |
| Parallel: equal slopes; perpendicular: m₁ × m₂ = −1 | Geometry, Algebra 1 unit 4 | Forgetting to flip and change the sign |
| Distance d = √[(x₂ − x₁)² + (y₂ − y₁)²] | Length of a segment | Forgetting the square root |
| Midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) | Centre of a segment; circle centres | Subtracting instead of adding |
| Section formula ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)) | CBSE Class 10 | Swapping m and n |
5. Sequences and series
| Formula | Use it when | The mistake |
|---|---|---|
| Arithmetic nth term: aₙ = a₁ + (n − 1)d | A constant difference | Using n instead of n − 1 |
| Arithmetic sum: Sₙ = n/2 [2a₁ + (n − 1)d] = n/2 (a₁ + aₙ) | Adding the first n terms | Mixing the two forms |
| Geometric nth term: aₙ = a₁ rⁿ⁻¹ | A constant ratio | rⁿ instead of rⁿ⁻¹ |
| Geometric sum: Sₙ = a₁(1 − rⁿ)/(1 − r), r ≠ 1 | Adding the first n terms | Sign of 1 − r |
| Infinite geometric sum: S = a₁/(1 − r), | r | < 1 |
6. Exponentials and logarithms (Algebra 2, Class 11)
| Formula | Use it when | The mistake |
|---|---|---|
| Growth and decay: y = a(1 + r)ᵗ, y = a(1 − r)ᵗ | Population, prices, radioactivity | Using r as a percent instead of a decimal |
| Compound interest: A = P(1 + r/n)ⁿᵗ; continuous A = Peʳᵗ | Finance problems | n for monthly is 12, not 1 |
| Simple interest: I = Prt | CBSE Class 7 to 8; comparison with compound | t in years |
| logₐ x = y means aʸ = x | Converting between forms | Reading the base and the argument backwards |
| log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x | Solving exponential equations; simplifying | log(x + y) is not log x + log y |
| Change of base: logₐ x = log x / log a | A base the calculator does not have | Dividing the wrong way |
| ln e = 1, log 1 = 0, log₁₀ 10 = 1 | Checking |
7. Percent, ratio and the small formulas that recur
| Formula | Use it when | The mistake |
|---|---|---|
| Percent change = (new − old)/old × 100 | Any percentage question | Dividing by the new value |
| Discount price = P(1 − r); markup P(1 + r) | Shopping, tax | Taking the percent off twice |
| Average speed = total distance / total time | Motion problems | Averaging the two speeds |
| Work: 1/t = 1/t₁ + 1/t₂ | Two people or pipes working together | Adding the times |
| Direct variation y = kx; inverse y = k/x | "Proportional to" | Confusing the two |
| x | = a means x = a or x = −a | |
| Inequalities: multiplying or dividing by a negative reverses the sign | Every inequality | Not reversing |
How to learn them
Not as a list. Each formula on this page comes from a problem type, and the way to make it permanent is to meet the problem, derive or apply the formula, and use it ten times in a week; the sheet is for checking, not for chanting. Three that should be derived once rather than memorised: (a + b)² from (a + b)(a + b); the quadratic formula from completing the square; and the arithmetic sum from pairing the first and last terms. A student who has derived them does not forget them. Where the formulas sit in each course is in the Algebra 1 curriculum and Algebra 2 syllabus guides; what they are for outside the classroom is in algebra in everyday life.
Download the algebra formulas sheet (PDF)
Where to get help
A 1-on-1 online class with a tutor who holds a master's in mathematics: the formulas taught from the problems that need them, derived once on the shared whiteboard, and tested weekly until automatic; Grade 8 to Algebra 2, CBSE Class 8 to 11, GCSE, IGCSE and IB. The recording and notes after every class, the same tutor all year, US$10 to $13 an hour. The Algebra 1 and Algebra 2 pages have the units; the free 30-minute first class is a real lesson on this week's topic.
Questions parents ask
1What are the basic algebra formulas?
2What are the algebraic identities?
3How do I memorise algebra formulas?
4Are the formulas the same in CBSE, GCSE and US Algebra?
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