Algebra formulas

The algebra formulas you reach for most: solving quadratics (and reading the discriminant), expanding brackets with the binomial theorem, the logarithm and exponent rules, and the exponential models behind population growth, radioactive decay and percentage change.

Printable cheat sheet Formula finder

Quadratic formula

x = (−b ± √(b² − 4ac))/(2a)

Solves any quadratic equation $ax² + bx + c = 0$ from its three coefficients, including the cases with one root or complex roots.

Discriminant

D = b² − 4ac

Tells you, before solving, whether a quadratic has two real roots, one repeated root or no real roots.

Vertex of a parabola

h = −b/(2a), k = c − b²/(4a)

The turning point (h, k) of $y = ax² + bx + c$ : its minimum when a > 0 and its maximum when a < 0.

Difference of two squares

a² − b² = (a + b)(a − b)

Any square minus another square factorises into a sum times a difference.

Binomial theorem

(a + b)ⁿ = ∑[k = 0..n] C(n, k)a^(n − k)b^k

Expands $(a + b)ⁿ$ without multiplying out brackets; the coefficients are the numbers in Pascal's triangle.

Definition of a logarithm

y = log_b x ⇔ b^y = x

A logarithm is an exponent: $log_b x$ is the power you raise b to in order to get x.

Logarithm product rule

log_b (MN) = log_b M + log_b N

The log of a product is the sum of the logs, which turns multiplication into addition.

Logarithm quotient rule

log_b (M/N) = log_b M − log_b N

The log of a quotient is the difference of the logs.

Logarithm power rule

log_b (M^p) = plog_b M

An exponent inside a log can come out in front as a multiplier, which is how you solve for an unknown exponent.

Change of base formula

log_b x = (ln x)/(ln b) = (log x)/(log b)

Works out a logarithm in any base using the ln or log₁₀ button on a calculator.

Laws of exponents

a^m·aⁿ = a^(m + n)

Multiplying powers of the same base adds the exponents; the other index laws follow from this one.

Exponential growth and decay

N = N₀e^(kt)

Models any quantity that changes at a rate proportional to its size: populations, continuous interest, cooling, decay.

Half-life formula

N = N₀(1/2)^(t/T)

How much of a decaying substance is left after time t, given its half-life T.

Percentage change

p = (V₂ − V₁)/(V₁)×100%

The change from an old value to a new one, as a percentage of the old value.