Algebra Formulas

a^-x = 1 / a^x

a^1/x = ^x√a

a^-1/x = 1 / ^x√a

(a.b)^x = a^x . b^x

(a^x)^y = a^x.y

aˣ⁺ʸ = aˣ . aʸ
aˣ⁻ʸ = aˣ / aʸ

e.g.: a^2.5 = a² . a^0.5
e.g.: a^0.25 = a¹ / a^0.75

xⁿ/yⁿ + 1 = (xⁿ + yⁿ) / yⁿ

(x - a) / A = x/A - a/A

(x² - a²) = (x - a).(x + a)

Simplify: (x + b)/(x² - a²)
Note: (x² - a²) = (x - a).(x + a)

(x + b)/(x² - a²) = (x + b) / [(x - a).(x + a)]
(x + b) / [(x - a).(x + a)] = A / (x - a) + B / (x + a) {where A & B are unknown}
(x + b) = A.[(x - a).(x + a)] / (x - a) + B.[(x - a).(x + a)] / (x + a)
(x + b) = A.(x + a) + B.(x - a)
Find A:
set x = a
(x + b) = A.(a + a) + B.(a - a)
(x + b) = 2.a.A
A = (x + b) / 2.a
Find B:
set x = -a
(x + b) = A.(-a + a) + B.(-a - a)
(x + b) = B.(-a - a)
B = (x + b) / 2.-a
(x + b)/(x² - a²) = (x + b)/[2.a.(x - a)] + (x + b)/[2.-a.(x + a)]
cancel (x + b):
1/(x² - a²) = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]

Continue in order to prove the above:
1/[(x - a).(x + a)] = 1/[2.a.(x - a)] - 1/[2.a.(x + a)]
1 = [(x - a).(x + a)]/[2.a.(x - a)] - [(x - a).(x + a)]/[2.a.(x + a)]
1 = (x + a)/(2.a) - (x - a)/(2.a)
1 = [1/(2.a)] . [(x + a) - (x - a)]
2.a = [(x + a) - (x - a)]
2.a = x + a - x + a
2.a = a + a

Factorial: e.g. 5! = 5x4x3x2x1

(a+b)ⁿ = aⁿ + ⁿC₁.a^(n-1).b + ⁿC₂.a^(n-2).b2 + ..... + ⁿC_r.a^(n-r).b^r + ..... bⁿ
Where:
ⁿC_r = n! / (n-1)!.r!
ⁿC₁ = n
ⁿC₂ = n! / (n-2)!.2! = n.(n-1) / 2!
ⁿC₃ = n! / (n-3)!.3! = n.(n-1)(n-2) / 3!
etc.
first few terms are as follows;
(a+b)ⁿ = aⁿ + n.a^(n-1).b + n(n-1).a^(n-2).b²/2! + n(n-1)(n-2).a^(n-3).b³/3! + ..... + bⁿ

If; 0 = ax² + bx + c
then; x = -b ± (b² - 4.a.c)^½ / 2.a

I = P.p.n

P_n = I + P

I = P_n - P

P_n = P.(1 + p/q)^n.q

m = P_n / n.q

V = P_n / (1+p/q)^n.q

Discount(simple) = P_n(simple) - V

Discount(compound) = P_n(compound) - V

n^th term = 1^st term + d.(n-1)

Σn terms = n.[2 . 1^st term.(n-1)] / 2

n^th term = 1^st term . r^(n-1))

Σn terms = 1^st term . (1 - rⁿ) / (1 - r)      [r < 1]

Σn terms = 1^st term . (rⁿ - 1) / (r - 1)      [r > 1]

e = 1 + 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ..... = 2.71828182845905

e^x = 1 + x¹/1 + x²/2 + x³/3 + x⁴/4 + x⁵/5 + x⁶/6 + x⁷/7 + ....

e^-x = 1 - x¹/1 + x²/2 - x³/3 + x⁴/4 - x⁵/5 + x⁶/6 - x⁷/7 + ....

y = Log_base(base^y)

Log x = Log_base(x)
Anti-Log x = base^x

log_base(x) = log_a(x) / log_a(base)
Where 'a' can be any number; 1, 2.71828182845905, 10, etc.

base^(a+b+c) = base^(a).base^(b).base^(c)
e.g.: exp(a+b+c) = exp(a).exp(b).exp(c)

Log_base(x) - Log_base(y) = log₁₀(base).x/y

Log_base(x^a) = a.Log_base(x)

Log_base(x) - Log_base(y) = z
Same as: x/y = base^z

1/exp(x) = exp(-x)

e^ln(x) = x

Xn = Y
n = Log(Y) / Log(X)

2ᴺᴰ Order

a₁.w + b₁.x = 0
a₂.w + b₂.x = 0

|a₁,b₁| = a₁.b₂ - a₂.b₁
|a₂,b₂|

To solve a 2ᴺᴰ Order equation you perform the calculation as shown above

3ᴿᴰ Order

a₁.w + b₁.x + c₁.y = 0
a₂.w + b₂.x + c₂.y = 0
a₃.w + b₃.x + c₃.y = 0

|a₁,b₁,c₁| = a₁.|b₂.c₂| - a₂.|b₁.c₁| + a₃.|b₁.c₁|
|a₂,b₂,c₂|         |b₃.c₃|       |b₃.c₃|         |b₂.c₂|
|a₃,b₃,c₃|

To solve a 3ᴿᴰ Order equation you convert to 2ᴺᴰ Order equations as shown above then solve 2ᴺᴰ Order equations

4ᵀᴴ Order

a₁.w + b₁.x + c₁.y + d₁.z = 0
a₂.w + b₂.x + c₂.y + d₂.z = 0
a₃.w + b₃.x + c₃.y + d₃.z = 0
a₄.w + b₄.x + c₄.y + d₄.z = 0

|a₁,b₁,c₁.d₁| = a₁.|b₂.c₂.d₂| - a₂.|b₁.c₁.d₁| + a₃.|b₁.c₁.d₁| - a₄.|b₁.c₁.d₁|
|a₂,b₂,c₂,d₂|        |b₃.c₃.d₃|       |b₃.c₃.d₃|         |b₂.c₂.d₂|        |b₂.c₂.d₂|
|a₃,b₃,c₃,d₃|        |b₄.c₄.d₄|       |b₄.c₄.d₄|         |b₄.c₄.d₄|        |b₃.c₃.d₃|
|a₄,b₄,c₄,d₄|

To solve a 4�€ᴴ Order equation you convert to 3ᴿᴰ Order equations as shown above, then convert to 2ᴺᴰ Order equations then solve 2ᴺᴰ Order equations