Complete reference for 500+ mathematical formulas with step-by-step examples and real-world applications
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x = (-b ± √(b² - 4ac)) / (2a)
Solves any quadratic equation of the form ax² + bx + c = 0
Δ = b² - 4ac
Determines the nature of roots in a quadratic equation
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Calculates the distance between two points in a coordinate plane
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Finds the exact middle point between two coordinates
m = (y₂ - y₁) / (x₂ - x₁)
Calculates the steepness or inclination of a line
aᵐ × aⁿ = aᵐ⁺ⁿ
When multiplying powers with the same base, add the exponents
aᵐ / aⁿ = aᵐ⁻ⁿ
When dividing powers with the same base, subtract the exponents
(aᵐ)ⁿ = aᵐⁿ
When raising a power to another power, multiply the exponents
(ab)ⁿ = aⁿbⁿ
When raising a product to a power, raise each factor to that power
(a/b)ⁿ = aⁿ/bⁿ
When raising a quotient to a power, raise numerator and denominator separately
a⁰ = 1
Any non-zero number raised to the power of 0 equals 1
a⁻ⁿ = 1/aⁿ
A negative exponent indicates the reciprocal
a^(m/n) = ⁿ√(aᵐ)
A fractional exponent represents a root
logₐ(x) = y ⟺ aʸ = x
Logarithm Definition for logarithmic expressions
logₐ(xy) = logₐ(x) + logₐ(y)
Product Rule (Logarithm) for logarithmic expressions
logₐ(x/y) = logₐ(x) - logₐ(y)
Quotient Rule (Logarithm) for logarithmic expressions
logₐ(xⁿ) = n·logₐ(x)
Power Rule (Logarithm) for logarithmic expressions
logₐ(x) = logᵦ(x) / logᵦ(a)
Change of Base Formula for logarithmic expressions
ln(x) = logₑ(x)
Natural Logarithm for logarithmic expressions
log(x) = log₁₀(x)
Common Logarithm for logarithmic expressions
a² + b² = c²
Relates the three sides of a right triangle
A = ½bh
Calculates the area of any triangle using base and height
A = πr²
Calculates the area enclosed by a circle
C = 2πr
Calculates the perimeter (distance around) a circle
A = l × w
Area of a rectangle
A = s²
Area of Square
A = bh
Area of Parallelogram
A = ½(b₁ + b₂)h
Area of Trapezoid
P = 2(l + w)
Perimeter of Rectangle
P = 4s
Perimeter of Square
V = s³
Volume of Cube
V = (4/3)πr³
Volume of Sphere
V = πr²h
Volume of Cylinder
V = (1/3)πr²h
Volume of Cone
V = lwh
Volume of Rectangular Prism
SA = 4πr²
Surface Area of Sphere
SA = 2πr² + 2πrh
Surface Area of Cylinder
SA = 6s²
Surface Area of Cube
sin(θ) = opposite / hypotenuse
Defines sine as ratio in a right triangle
cos(θ) = adjacent / hypotenuse
Defines cosine as ratio in a right triangle
tan(θ) = opposite / adjacent
Defines tangent as ratio in a right triangle
sin²(θ) + cos²(θ) = 1
Fundamental identity relating sine and cosine
tan(θ) = sin(θ)/cos(θ)
Tangent Identity
sin(2θ) = 2sin(θ)cos(θ)
Double Angle (Sine)
cos(2θ) = cos²(θ) - sin²(θ)
Double Angle (Cosine)
a/sin(A) = b/sin(B) = c/sin(C)
Law of Sines
c² = a² + b² - 2ab·cos(C)
Law of Cosines
sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
Sum of Angles (Sine)
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
Sum of Angles (Cosine)
d/dx[xⁿ] = n·xⁿ⁻¹
Differentiates power functions
d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Differentiates product of two functions
d/dx[f(g(x))] = f'(g(x))·g'(x)
Differentiates composite functions
d/dx[f/g] = (f'g - fg')/g²
Quotient Rule
d/dx[sin(x)] = cos(x)
Derivative of sin(x)
d/dx[cos(x)] = -sin(x)
Derivative of cos(x)
d/dx[tan(x)] = sec²(x)
Derivative of tan(x)
d/dx[eˣ] = eˣ
Derivative of eˣ
d/dx[ln(x)] = 1/x
Derivative of ln(x)
∫xⁿdx = xⁿ⁺¹/(n+1) + C
Power Rule (Integral)
∫sin(x)dx = -cos(x) + C
Integral of sin(x)
∫cos(x)dx = sin(x) + C
Integral of cos(x)
∫eˣdx = eˣ + C
Integral of eˣ
∫(1/x)dx = ln|x| + C
Integral of 1/x
μ = (Σx) / n
Calculates the average of a dataset
σ = √[Σ(x-μ)² / n]
Measures spread of data from the mean
Formula here
Median
Formula here
Mode
Formula here
Variance
Formula here
Range
Formula here
Probability
Formula here
Permutation
Formula here
Combination
d = v × t
Distance = Speed × Time
a = Δv / Δt
Acceleration
F = ma
Force (Newton's 2nd Law)
p = mv
Momentum
KE = ½mv²
Kinetic Energy
PE = mgh
Potential Energy
W = F × d
Work
P = W / t
Power
V = IR
Ohm's Law
P = VI
Electric Power
R = ρL/A
Resistance
v = fλ
Wave Speed
f = 1/T
Frequency-Period
M = mass / moles
Molar Mass Calculation
M = n / V
Molarity
PV = nRT
Ideal Gas Law
ρ = m / V
Density
% = (mass of element / mass of compound) × 100
Percent Composition
pH = -log[H⁺]
pH Formula
M₁V₁ = M₂V₂
Dilution Formula
P₁V₁ = P₂V₂
Boyle's Law
V₁/T₁ = V₂/T₂
Charles' Law
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