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Algebra Formulas
Every essential algebra formula: quadratics, factoring, exponents, logarithms, sequences, and lines.
Quadratic equations
| Quadratic formula | x = (−b ± √(b² − 4ac)) / 2a | Solves ax² + bx + c = 0 |
| Discriminant | Δ = b² − 4ac | >0: two real roots; =0: one; <0: none |
| Sum / product of roots | x₁ + x₂ = −b/a, x₁·x₂ = c/a | Vieta's formulas |
Factoring patterns
| Difference of squares | a² − b² = (a − b)(a + b) | |
| Perfect square | a² ± 2ab + b² = (a ± b)² | |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) | |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) | |
Exponents
| Product | aᵐ · aⁿ = aᵐ⁺ⁿ | |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | |
| Negative exponent | a⁻ⁿ = 1 / aⁿ | |
| Zero exponent | a⁰ = 1 (a ≠ 0) | |
| Fractional exponent | a^(m/n) = ⁿ√(aᵐ) | |
Logarithms
| Product rule | log(ab) = log a + log b | |
| Quotient rule | log(a/b) = log a − log b | |
| Power rule | log(aⁿ) = n · log a | |
| Change of base | log_b(a) = log(a) / log(b) | |
| Inverse | b^(log_b x) = x | |
Sequences
| Arithmetic, nth term | aₙ = a₁ + (n−1)d | d = common difference |
| Arithmetic sum | Sₙ = n/2 · (2a₁ + (n−1)d) | |
| Geometric, nth term | aₙ = a₁ · rⁿ⁻¹ | r = common ratio |
| Geometric sum | Sₙ = a₁(1 − rⁿ) / (1 − r) | r ≠ 1 |
Lines
| Slope | m = (y₂ − y₁) / (x₂ − x₁) | |
| Point–slope | y − y₁ = m(x − x₁) | |
| Slope–intercept | y = mx + b | b = y-intercept |
| Distance | d = √((x₂−x₁)² + (y₂−y₁)²) | |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) | |