Algebra Formulas
A copy-able sheet — quadratic, special products, factoring, exponent, and logarithm rules.
Click any formula to copy it.
The quadratic formula, x = (−b ± √(b² − 4ac)) / (2a), solves any ax² + bx + c = 0. This sheet also collects the special products, factoring identities, exponent laws (aᵐ · aⁿ = aᵐ⁺ⁿ), and logarithm rules (logₐ(xy) = logₐx + logₐy) you reach for most in algebra.
One sheet for the core of algebra
Most of school and early-college algebra runs on a small set of identities: the quadratic formula, a handful of special products and their reverse (factoring), the laws of exponents, and the rules for logarithms. The panel above groups them so you can filter to the family you need and copy any line straight into your work.
How the families connect
Special products and factoring are the same identities read in opposite directions — expanding (a + b)² gives a² + 2ab + b², and recognising a² − b² lets you factor it back to (a + b)(a − b). Exponent and logarithm rules are likewise mirror images: a logarithm undoes a power, which is why logₐ(xⁿ) = n logₐx follows from (aᵐ)ⁿ = aᵐⁿ.
Quadratic equations
For any equation of the form ax² + bx + c = 0 with a ≠ 0.
| Quantity | Formula | What it gives |
|---|---|---|
| Roots | x = (−b ± √(b² − 4ac)) / (2a) | The two solutions of ax² + bx + c = 0 |
| Discriminant | Δ = b² − 4ac | Number of real roots: 2 if Δ > 0, 1 if Δ = 0, 0 if Δ < 0 |
| Vertex (axis) | x = −b / (2a) | The x-value of the parabola’s turning point |
Laws of exponents
Hold for real bases a, b and exponents m, n (with a ≠ 0 where a appears in a denominator).
| Rule | Identity |
|---|---|
| Product of powers | aᵐ · aⁿ = aᵐ⁺ⁿ |
| Quotient of powers | aᵐ / aⁿ = aᵐ⁻ⁿ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ |
| Power of a product | (ab)ⁿ = aⁿbⁿ |
| Zero exponent | a⁰ = 1 (a ≠ 0) |
| Negative exponent | a⁻ⁿ = 1 / aⁿ |
| Fractional exponent | a^(1/n) = ⁿ√a |
Logarithm rules
For base a > 0 with a ≠ 1, and positive arguments x, y.
| Rule | Identity |
|---|---|
| Product | logₐ(xy) = logₐx + logₐy |
| Quotient | logₐ(x/y) = logₐx − logₐy |
| Power | logₐ(xⁿ) = n logₐx |
| Change of base | logₐx = ln x / ln a |
| Log of one | logₐ1 = 0 |
| Log of the base | logₐa = 1 |
The signs that trip people up
The sum of cubes keeps a minus in the middle: a³ + b³ = (a + b)(a² − ab + b²), while the difference of cubes flips both outer signs to a plus in the middle: a³ − b³ = (a − b)(a² + ab + b²). The cube of a difference alternates signs term by term: (a − b)³ = a³ − 3a²b + 3ab² − b³. When a factoring attempt fails, a wrong middle sign is usually why.