Math Curves & Lines   Imaginary math   Trigonometry   Calculus   Area & Volume   Series ```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) ``` Home Area, Volume Atomic Mass Black Body Radiation Boolean Algebra Calculus Capacitor Center of Mass Carnot Cycle Charge Chemistry   Elements   Reactions Circuits Complex numbers Constants Curves, lines deciBell Density Electronics Elements Flow in fluids Fourier's Law Gases Gravitation Greek Alphabet Horizon Distance Interest Magnetics Math   Trig Math, complex Maxwell's Eq's Motion Newton's Laws Octal/Hex Codes Orbital Mechanics Particles Parts, Analog IC   Digital IC   Discrete Pendulum Planets Pressure Prime Numbers Questions Radiation Refraction Relativistic Motion Resistance, Resistivity Rotation Series SI (metric) prefixes Skin Effect Specific Heat Springs Stellar magnitude Thermal Thermal Conductivity Thermal Expansion Thermodynamics Trigonometry Units, Conversions Vectors Volume, Area Water Wave Motion Wire, Cu   Al   metric Young's Modulus