Algebra cheat sheet

All 14 formulas on one page. Print it or save it as a PDF; it fits on 3 sides of A4 or Letter.

Quadratic formula

x = (−b ± √(b² − 4ac))/(2a)

$x$ the solutions (roots) of the equation: the x-values where $ax² + bx + c$ equals 0
$a$ the coefficient of $x^{2}$ ; it must not be 0, or the equation is not quadratic
$b$ the coefficient of $x$ (use 0 if there is no $x$ term)
$c$ the constant term (use 0 if there is none)
$\pm$ "plus or minus": take + for one root and − for the other

D = b² − 4ac

h = −b/(2a), k = c − b²/(4a)

Vertex form

y = a(x − h)² + k

k from h

k = ah² + bh + c

$h$ the x-coordinate of the vertex; the line $x = h$ is the axis of symmetry
$k$ the y-coordinate of the vertex: the minimum or maximum value of y
$a$ coefficient of $x^{2}$ (not 0); its sign says whether the vertex is a minimum or a maximum
$b$ coefficient of $x$
$c$ constant term

a² − b² = (a + b)(a − b)

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

General term (the calculator)

coefficient of x^k in (ax + b)ⁿ = C(n, k)a^kb^(n − k)

y = log_b x ⇔ b^y = x

Natural log

ln x = logₑ x

Common log

log x = log₁₀ x

log_b (MN) = log_b M + log_b N

log_b (M/N) = log_b M − log_b N

log_b (M^p) = plog_b M

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

a^m·aⁿ = a^(m + n)

Quotient

a^m/(aⁿ) = a^(m − n)

Power of a power

(a^m)ⁿ = a^(mn)

Zero and negative

a⁰ = 1, a^( −n) = 1/(aⁿ)

Fractional

a^(1/n) = √a

N = N₀e^(kt)

Doubling time

t₂ = (ln 2)/k

Growth factor form

N = N₀(1 + r)^t, k = ln (1 + r)

$N$ the amount at time t
$N_{0}$ the starting amount (at t = 0)
$k$ the continuous growth rate per unit time; positive for growth, negative for decay
$t$ the time elapsed, in the same units as k
$e$ Euler's number, about 2.71828

N = N₀(1/2)^(t/T)

Decay constant

N = N₀e^( −λt), λ = (ln 2)/T

p = (V₂ − V₁)/(V₁)×100%

Reverse percentage

V₁ = V₂/(1 + p/100)

letx.io · Algebra cheat sheet · 2026-10-31