Essential formulas for algebra, geometry, calculus, physics, and chemistry
x = [-b ± √(b² - 4ac)] / 2a
aᵐ × aⁿ = aᵐ⁺ⁿ
(aᵐ)ⁿ = aᵐⁿ
aᵐ / aⁿ = aᵐ⁻ⁿ
logₐ(xy) = logₐx + logₐy
logₐ(x/y) = logₐx - logₐy
logₐ(xⁿ) = n·logₐx
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
Square: P = 4s
Rectangle: P = 2(l + w)
Circle: C = 2πr
Square: A = s²
Rectangle: A = l × w
Triangle: A = ½ × b × h
Circle: A = πr²
Trapezoid: A = ½(b₁ + b₂)h
Cube: V = s³
Sphere: V = (4/3)πr³
Cylinder: V = πr²h
Cone: V = (1/3)πr²h
a² + b² = c²
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ - sin²θ
tan 2θ = (2 tan θ)/(1 - tan²θ)
a/sin A = b/sin B = c/sin C
c² = a² + b² - 2ab cos C
d/dx (xⁿ) = n·xⁿ⁻¹
d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
d/dx (uv) = u·dv/dx + v·du/dx
dy/dx = dy/du × du/dx
v = u + at
s = ut + ½at²
v² = u² + 2as
F = ma (Force = mass × acceleration)
p = mv (Momentum)
W = F × d (Work)
KE = ½mv² (Kinetic Energy)
PE = mgh (Potential Energy)
V = IR (Ohm's Law)
P = IV (Power)
F = qE (Electric Force)
n = mass / molar mass
n = volume / 22.4 L (STP)
n = C × V (Concentration)
PV = nRT
pH = -log[H⁺]
pOH = -log[OH⁻]
pH + pOH = 14
Rate = k[A]ᵃ[B]ᵇ
k = Ae⁻ᴱᵃ/ᴿᵀ
Mean = Σx / n
σ = √[Σ(x - μ)² / N]
P(A) = favorable outcomes / total outcomes
P(A∩B) = P(A) × P(B) (independent)
P(X=k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ
Z = (X - μ) / σ