Math
quadratic equation: to solve ax² + bx + c = 0 x = [–b ± √(b²–4ac)] / 2a x = [–b ± √(b²–4ac)] / 2a Discriminant Δ = √(b²–4ac) Δ = 0, one real root –b/2a Δ < 0, no real roots, two comples roots Δ > 0, two real roots completing the square x² + bx + c = a(x – h)² + k where h = –b/2 and k = c – (b²/4) ax² + bx + c = a(x – h)² + k where h = –b/2a and k = c – (b²/4a) distance between two points d = √(Δx² + Δy² + Δz²) Distance point to line line is ax + by + c = 0 point is x₀, y₀ distance is |ax₀ + by₀ + c| / √(a² + b²) Binomial theorem: (Pascal's Triangle) Expansion of (x + y)ᴺ (x + y)² = x² + 2xy + y² (x – y)² = x² – 2xy + y² (x + y)³ = x³ + 3x²y + 3xy² + y³ (x – y)³ = x³ – 3x²y + 3xy² – y³ (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ 5: 1 5 10 10 5 1 6: 1 6 15 20 15 6 1 7: 1 7 21 35 35 21 7 1 8: 1 8 28 56 70 56 28 8 1 9: 1 9 36 84 126 126 84 36 9 1 10: 1 10 45 120 210 252 210 120 45 10 1 factors to factor x² + bx + c [x – (½)(–b+√(b²–4c))] [x – (½)(–b–√(b²–4c))] a² – b² = (a + b)(a – b) a² + b² = (a + bi)(a – bi) a³ + b³ = (a + b)(a² – ab + b²) a³ – b³ = (a – b)(a² + ab + b²) a⁴ – b⁴ = (a – b)(a³ + a²b + ab² + b³) a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴) Mean, average, etc of numbers Mean (average) of x, y, z = (1/3)(x+y+z) Geometric mean = ∛(xyz) Harmonic mean = 3(1/((1/x)+(1/y)+(1/z))) Median: the value separating the higher half of a data sample from the lower half. Average value of a function between a and b is AV(a,b) = (1/(b–a)) ∫ f(x) dx between a and b Center of Gravity in one dimension, for a object of uniform thickness: sum of moments divided by sum of areas, where moments are areas x distances. Exponents & Logs (k^a)(k^b) = k^(a+b) (k^a)^b = K^ab (hk)^a = (h^a)(k^a) h(k^a) = k^(–a) = 1 / k^a k^1 = k k^0 = 1 log ab = log a + log b log a/b = log a – log b log aᴺ = N log a x^logy = y^logx log(base a) x = [ log(base b) x ] / [ log(base b) a ] log(base k) k = 1 log 10 = 1 ln e = 1 10^(log k) = k e^(ln k) = k log 1 = 0 roots √(a + b√c) = √d + √e e = [a ±√(a²–b²c)] / 2 d = [a ∓√(a²–b²c)] / 2 √(3 + 2√2) = 1 + √2 Repeating Decimals x = 0.77777 ... 10x = 7.7777 ... 10x – x = 9x = 7 x = 7/9 x = 0.4777777 x = 0.4 + 0.07777 let y = 0.77777 x = 0.4 + (y/10) or x = (4/10) + (y/10) now convert 0.77777 into a fraction y = 0.77777 .. 10y = 7.77777 ... 10y – y = 7 9y = 7 y = 7/9 x = (4/10) + (7/90) x = (36/90) + (7/90) x = 43/90 x = 0.8363636 ... y = 0.3636363 ... x = (8/10) + (y/10) 100y = 36.363636 ... 100y – y = 36 y = 36/99 x = (8/10) + (36/990) x = (792/990) + (36/990) x = 828/990 = 46/55 Trapazoidal rule Trapazoidal rule is used to approx. area under a curve A = (b–a)(1/2)(f(a)–f(b) for more that 2 intervals A = Σ ( (1/2)(f(Xκ-₁) – f(Xκ))ΔXκ between 1 and N N is number of pairs – 1 A = (Δx/2)( f(x₀) + 2f(x₁) + 2f(x₂) + 2f(x₃) + ... + f(xᵣ) ) Δx = (xᵣ–x₀)/N (r subscript = N) Inequalities These do not change the direction of the inequality Add (or subtract) a number from both sides Multiply (or divide) both sides by a positive number Simplify a side These DO change the direction of the inequality Multiply (or divide) both sides by a negative number Swapping left and right hand sides Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always n egative) |
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