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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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