📑 Reference library

Math formulas to know cold

Algebra & lines

• Slope = (y₂ − y₁) / (x₂ − x₁)
• Slope-intercept form: y = mx + b  (m = slope, b = y-intercept)
• Point-slope form: y − y₁ = m(x − x₁)
• Distance between points: √[(x₂ − x₁)² + (y₂ − y₁)²]
• Midpoint: ((x₁ + x₂)/2 , (y₁ + y₂)/2)
• No solution = parallel lines (same slope, different intercept). Infinite solutions = the same line.

Quadratics & exponents

• Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
• Vertex: x = −b / (2a); in a(x − h)² + k the vertex is (h, k)
• Sum of roots = −b/a  ·  Product of roots = c/a
• Exponents: xᵃ · xᵇ = xᵃ⁺ᵇ  ·  xᵃ / xᵇ = xᵃ⁻ᵇ  ·  (xᵃ)ᵇ = xᵃᵇ  ·  x⁰ = 1  ·  x⁻ᵃ = 1/xᵃ
• Radicals: √x = x^(1/2)

Percent, rates & statistics

• Percent change = (new − old) / old × 100  (always divide by the ORIGINAL)
• Discount: pay (100 − discount)% of the original
• Mean = sum ÷ count  →  sum = mean × count
• Median = middle value in order; Mode = most frequent
• Probability = favorable outcomes ÷ total outcomes
• Exponential growth: final = start × (rate)^(time ÷ period)

Geometry

• Pythagorean theorem: a² + b² = c². Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25
• Circle: area = πr², circumference = 2πr
• Arc length = (angle/360) × 2πr  ·  Sector area = (angle/360) × πr²
• Triangle: area = ½ × base × height; angles sum to 180°
• Rectangle: area = lw, perimeter = 2(l + w)
• Cylinder volume = πr²h  ·  Box volume = lwh  ·  Sphere volume = (4/3)πr³
• Similar figures: matching sides are proportional (find the scale factor)

Trigonometry

• SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj
• Radians: 180° = π radians (multiply degrees by π/180)
• Special angles: 30-60-90 sides are x, x√3, 2x; 45-45-90 sides are x, x, x√2