Basic Algebra Formulas
7 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Algebra has a small set of formulas that appear again and again. Knowing them by heart saves time and prevents mistakes. This cheat sheet groups them by purpose, with an example of each and a few mental-math tricks that use them.
Algebraic identities
| Identity | Example |
|---|---|
| (a + b)² = a² + 2ab + b² | (x + 4)² = x² + 8x + 16 |
| (a − b)² = a² − 2ab + b² | (x − 3)² = x² − 6x + 9 |
| a² − b² = (a + b)(a − b) | x² − 25 = (x + 5)(x − 5) |
| (a + b)³ = a³ + 3a²b + 3ab² + b³ | (x + 1)³ = x³ + 3x² + 3x + 1 |
| a³ + b³ = (a + b)(a² − ab + b²) | x³ + 8 = (x + 2)(x² − 2x + 4) |
Mental-math tricks from the identities
- Squaring near a round number: 47² = (50 − 3)² = 2500 − 300 + 9 = 2209.
- Products that straddle a number: 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9991.
Exponent rules
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 and a⁻ⁿ = 1/aⁿ
More in exponents and powers explained.
Quadratic formula
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. The vertex of the parabola is at x = −b ÷ 2a. Solve any example with the quadratic equation solver.
Coordinate formulas
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
- Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
- Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
- Line: y = mx + b
Example: between (1, 2) and (4, 6) the distance is √(9 + 16) = 5 and the midpoint is (2.5, 4).
Sequences
Arithmetic: aₙ = a₁ + (n − 1)d. Geometric: aₙ = a₁rⁿ⁻¹. Try both in the sequence calculator.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Multiply (x + 3)(x − 5).
Show answer
Product: x² − 2x − 15
- Solve x² − 7x + 10 = 0.
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x₁: 5 · x₂: 2
- An arithmetic sequence starts at 5 with difference 3. Find the 20th term and the sum of 20 terms.
Show answer
Term 20 (aₙ): 62 · Sum of the first 20 terms: 670
Frequently asked questions
Which algebra formulas should I memorise?
The square and difference-of-squares identities, the exponent rules, the quadratic formula and the slope and distance formulas.
Is (a + b)² equal to a² + b²?
No. There is a middle term: (a + b)² = a² + 2ab + b².
How can identities speed up mental maths?
They let you square numbers near a round number and multiply pairs like 103 × 97 quickly.
Where can I practise?
Take the <a href="/quiz/algebra-1">Algebra 1 quiz</a>.
Try it yourself
Polynomial CalculatorAdd, subtract and multiply polynomials and evaluate the result at any x.Quadratic Equation SolverSolve ax² + bx + c = 0 with the discriminant, roots and vertex.Sequence CalculatorFind the n-th term, the sum and the first terms of an arithmetic or geometric sequence.Keep learning
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