Algebra
Equations, expressions, roots
Quadratic Formula
x =
−b ± b2 − 4ac2a
Solves any quadratic ax² + bx + c = 0. The discriminant Δ = b² − 4ac tells you how many real solutions exist.
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Difference of Two Squares
a2 − b2 = (a + b)(a − b)
Any expression of the form a² − b² factors into two conjugate binomials.
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Perfect Square Trinomial
(a + b)2 = a2 + 2ab + b2
Expanding the square of a binomial. Useful for completing the square.
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Sum of Cubes
a3 + b3 = (a + b)(a2 − ab + b2)
Factorises a sum of cubes. The difference of cubes reverses the sign: a³ − b³ = (a − b)(a² + ab + b²).
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Sum of Roots (Vieta's)
x1 + x2 =
−ba
,
x1x2 =
ca
For ax² + bx + c = 0, the sum and product of roots without solving explicitly.
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Completing the Square
ax2 + bx + c = a(x +
b2a)2
−
b2 − 4ac4a
Rewrites a quadratic in vertex form, useful for finding the minimum/maximum.
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Trigonometry
Angles, triangles, identities
Pythagorean Identity
sin2(θ) + cos2(θ) = 1
The most fundamental trig identity, valid for all angles θ. Useful for simplifying trig expressions.
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Sine Rule
asin A
=
bsin B
=
csin C
Relates each side of a triangle to the sine of its opposite angle. Use when you know 2 angles and 1 side, or 2 sides and a non-included angle.
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Cosine Rule
c2 = a2 + b2 − 2ab cos(C)
Generalises Pythagoras for any triangle. Use when you know 2 sides and the included angle, or all 3 sides.
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Area of a Triangle
A =
12
ab sin(C)
Area using two sides and the included angle. Works for any triangle when you know two sides and the angle between them.
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Double Angle — Sine
sin(2θ) = 2 sin(θ) cos(θ)
Express double angle in terms of single angles. Useful for integration and simplification.
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Double Angle — Cosine
cos(2θ) = cos2(θ) − sin2(θ)
Equivalently: cos(2θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ).
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Geometry
Areas, volumes, perimeters
Circle — Area & Circumference
A = πr2 , C = 2πr
Area enclosed by and circumference of a circle with radius r.
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Sphere — Volume & Surface Area
V =
43
πr3
,
A = 4πr2
Volume and surface area of a sphere of radius r.
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Cylinder
V = πr2h
Volume of a cylinder with base radius r and height h. Surface area = 2πr(r + h).
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Cone
V =
13
πr2h
Volume of a cone with base radius r and perpendicular height h.
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Trapezium Area
A =
(a + b)2
h
Area of a trapezium with parallel sides a and b, and perpendicular height h.
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Pythagoras' Theorem
a2 + b2 = c2
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
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Simultaneous Equations
Two equations, two unknowns
Cramer's Rule
x =
DxD
,
y =
DyD
Used internally to solve systems of linear equations. D is the determinant of the coefficient matrix; Dx and Dy replace the respective column with constants.
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Substitution Method
If y = mx + c, substitute into the other equation to eliminate y
Express one variable in terms of the other, then substitute. Especially useful when one equation is already solved for a variable.
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Elimination Method
Multiply equations so coefficients of one variable are equal, then subtract
Multiply one or both equations by a constant to align coefficients, then add/subtract to eliminate one variable.
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Calculus
Differentiation and integration rules
Power Rule — Differentiation
ddx
(xn) = n xn−1
The derivative of x to any power n. E.g. d/dx(x³) = 3x².
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Product Rule
(uv)′ = u′v + uv′
The derivative of a product of two functions. Differentiate each factor in turn while keeping the other fixed.
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Chain Rule
ddx
f(g(x)) = f′(g(x)) · g′(x)
Differentiate a composite function by multiplying the outer derivative (at the inner value) by the inner derivative.
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Power Rule — Integration
∫ xn dx =
xn+1n+1
+ C
Integral of x to a power (n ≠ −1). E.g. ∫x³ dx = x⁴/4 + C.
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Newton–Raphson Method
xn+1 = xn −
f(xn)f′(xn)
Iterative method for finding roots of a function. Start with an initial guess x0 and apply repeatedly until convergence.
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Fundamental Theorem of Calculus
ddx
∫ax f(t) dt = f(x)
Differentiation and integration are inverse operations. The derivative of an integral (from a constant to x) recovers the original function.
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Statistics & Data
Mean, spread, distributions
Mean (Average)
x̄ =
Σxn
Sum all values and divide by the count. The arithmetic mean is the most common measure of central tendency.
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Standard Deviation
σ =
Σ(x − x̄)2n
Measures how spread out data is from the mean. A higher σ means more spread. The sample standard deviation uses n−1.
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Normal Distribution (PDF)
f(x) =
1σ2π
e−½((x−μ)/σ)2
The bell-curve probability density. Symmetrical around mean μ with spread σ. Used in probability, hypothesis testing, and grading.
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Binomial Probability
P(X = r) =
nCr
pr(1−p)n−r
Probability of exactly r successes in n independent trials, each with probability p.
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Physics & Applied Maths
Essential equations from physics
Newton's Second Law
F = ma
Force (N) = mass (kg) × acceleration (m/s²). The net force on an object equals its rate of change of momentum.
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Kinetic Energy
KE =
12
mv2
Energy of an object in motion. Mass in kg, velocity in m/s, energy in joules.
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Gravitational PE
PE = mgh
Gravitational potential energy stored in an object at height h. Use g = 9.81 m/s².
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SUVAT — Distance
s = ut +
12
at2
Distance under constant acceleration. u = initial velocity, a = acceleration, t = time.
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Compound Interest
A = P(1 +
rn
)nt
Final amount A after t years at annual rate r, compounded n times per year with principal P.
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E = mc²
E = mc2
Mass-energy equivalence. c ≈ 3 × 10⁸ m/s. Shows that a small mass represents enormous energy.
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Financial & General
Percentages, growth, interest
Percentage Change
%Δ =
New − OldOld
× 100
Expresses a change as a percentage of the original value. Positive = increase, negative = decrease.
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Simple Interest
A = P(1 + rt)
Interest calculated only on the original principal P, at rate r per year for t years.
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Exponential Growth / Decay
N(t) = N0 ekt
k > 0 gives exponential growth, k < 0 gives decay (e.g. radioactive decay, population growth).
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Half-Life
t½ =
ln 2λ
Time for a quantity to reduce to half its initial value. λ is the decay constant. Used in nuclear physics and pharmacology.
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