Algebra Formulas

Quadratic Equation: The solution is given by the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Example: Solve 2x² + 3x - 5 = 0

x = (-3 ± √(3² - 4(2)(-5))) / 2(2)

x = (-3 ± √(9 + 40)) / 4

x = (-3 ± √49) / 4

x = (-3 ± 7) / 4

x = 1 or x = -2.5

Coordinate Geometry

Equation of a Line: y = mx + c, where m is the slope and c is the y-intercept.

Example: Find the equation of the line with slope m = 3 passing through the point (2, 5).

y - 5 = 3(x - 2)

y = 3x - 1

Trigonometry

Basic Trigonometric Functions:

sin θ = Opposite / Hypotenuse

cos θ = Adjacent / Hypotenuse

tan θ = Opposite / Adjacent

Cosine Rule

Cosine Rule for Triangles:

c² = a² + b² - 2ab cos(C)

Example: For a triangle with sides a = 5, b = 7, and angle C = 60°:

c² = 5² + 7² - 2(5)(7) cos(60°)

c² = 25 + 49 - 70(0.5)

c² = 25 + 49 - 35

c² = 39

c = √39 ≈ 6.24

Pythagorean Theorem

Pythagorean Theorem: In a right-angled triangle:

a² + b² = c²

Example: If a = 3 and b = 4, find the hypotenuse c.

3² + 4² = c²

9 + 16 = c²

25 = c²

c = √25 = 5

Set Theory

Union of Sets: A ∪ B

All elements in A or B

Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∪ B = {1, 2, 3, 4}

Intersection of Sets: A ∩ B

All elements common to A and B

Example: If A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}

Advanced Set Theory:

Difference: A - B

A = {1, 2, 3}, B = {2, 3, 4}, A - B = {1}

Symmetric Difference: A ∆ B

A ∆ B = (A - B) ∪ (B - A) = {1, 4}

Matrix Operations for Simultaneous Equations

For a system of 3 equations:

ax + by + cz = d

ex + fy + gz = h

ix + jy + kz = l

Solution using Cramer's Rule:

X = A⁻¹B

Where A is the coefficient matrix and B is the constants matrix.

Calculus

Chain Rule:

dy/dx = f'(g(x)) * g'(x)

Example: For y = sin(3x), the derivative is:

dy/dx = cos(3x) * 3 = 3cos(3x)

Advanced Differentiation

Product Rule:

d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Example: For u(x) = x² and v(x) = sin(x):

d/dx [x² sin(x)] = 2x sin(x) + x² cos(x)

Quotient Rule:

d/dx [u(x)/v(x)] = [v(x)u'(x) - u(x)v'(x)] / (v(x))²

Example: For u(x) = x and v(x) = cos(x):

dy/dx = [cos(x)(1) - x(-sin(x))] / cos²(x)

dy/dx = (cos(x) + xsin(x)) / cos²(x)

Statistics

Variance:

Var(X) = Σ(xᵢ - μ)² / n

Example: If X = {1, 2, 3}, μ = 2:

Var(X) = [(1-2)² + (2-2)² + (3-2)²] / 3

Var(X) = (1 + 0 + 1) / 3 = 2 / 3 ≈ 0.6667

Standard Deviation:

σ = √(Var(X))

Example: If Var(X) = 0.6667:

σ = √0.6667 ≈ 0.8165

Financial Formulas

Simple Interest:

SI = P × R × T / 100

Example: If P = 1000, R = 5%, T = 3 years:

SI = 1000 × 5 × 3 / 100 = 150

Compound Interest:

A = P(1 + r/n)^(nt)

Example: If P = 1000, r = 5%, n = 4, t = 2 years:

A = 1000(1 + 0.05/4)^(4×2) = 1000(1.0125)^8 ≈ 1105.22

Annuity Formula:

A = P × [(1 + r)^t - 1] / r

Example: If P = 1000, r = 0.05, t = 10 years:

A = 1000 × [(1 + 0.05)^10 - 1] / 0.05 ≈ 1000 × [1.6289 - 1] / 0.05 ≈ 12577.68

Factorial Formulas

Factorial: n! = n × (n - 1) × ... × 1

Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

Advanced Factorial Formula:

Γ(n) = (n - 1)! (Gamma function)

Example: Γ(5) = 4! = 4 × 3 × 2 × 1 = 24

Geometry Formulas

Area of a Circle:

A = πr²

Example: If r = 3:

A = π(3)² = 9π ≈ 28.27

Volume of a Cone:

V = (1/3)πr²h

Example: If r = 3 and h = 4:

V = (1/3)π(3)²(4) = 12π ≈ 37.70

Volume of a Pyramid:

V = (1/3)Bh

Example: If base area B = 25 and height h = 6:

V = (1/3)(25)(6) = 50