Ultimate math formulas
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Expansion and factorization
(a + b)^2 = a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3
(a + b)^n = \displaystyle \sum_{k = 0}^n \binom{n}{k} a^{n - k} b^k (Binomial theorem)
a^2 - b^2 = (a + b)(a - b)
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
a^n - b^n = (a - b) \displaystyle \sum_{k = 0}^{n - 1} a^{n - 1 - k} b^k
Quadratic formula
ax^2 + bx + c = 0
\implies x = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Exponent and logarithm rules
a^m a^n = a^{m + n}
(a^m)^n = a^{mn}
(ab)^n = a^n b^n
\log_a(MN) = \log_a M + \log_a N
\log_a(M^n) = n \log_a M
\log_a M = \displaystyle \frac{\log_b M}{\log_b a}
Fractions
\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\displaystyle \frac{a}{b} + \frac{c}{d} + \frac{e}{f} = \frac{adf + bcf + bde}{bdf}
Systems of linear equations
\begin{cases} ax + by = e \\ cx + dy = f \end{cases} \iff \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} e \\ f \end{pmatrix}
x = \displaystyle \frac{ed - bf}{ad - bc} = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}
y = \displaystyle \frac{af - ec}{ad - bc} = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}
Sequences
\displaystyle \sum_{k = 1}^n k = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}
\displaystyle \sum_{k = 1}^n \bigl( a + (k - 1)d \bigr) = a + (a + d) + \cdots + \bigl( a + (n - 1)d \bigr) = \frac{n \bigl( 2a + (n - 1)d \bigr)}{2}
\displaystyle \sum_{k = 1}^n ar^{k - 1} = a + ar + \cdots + ar^{n - 1} = \frac{a(1 - r^n)}{1 - r}
Derivatives
(x^n)' = nx^{n - 1}
Trigonometric identities
\sin^2 \theta + \cos^2 \theta = 1 (Pythagorean identity)
Addition formulas
\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
Double-angle formulas
\sin 2\theta = 2 \sin \theta \cos \theta
\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 1 - 2 \sin^2 \theta = 2 \cos^2 \theta - 1
Multiple-angle formulas
\sin n\theta = \displaystyle \sum_{k = 0}^{\left\lfloor \frac{n - 1}{2} \right\rfloor} (-1)^k \binom{n}{2k + 1} \cos^{n - (2k + 1)} \theta \sin^{2k + 1} \theta
\cos n\theta = \displaystyle \sum_{k = 0}^{\left\lfloor \frac{n}{2} \right\rfloor} (-1)^k \binom{n}{2k} \cos^{n - 2k} \theta \sin^{2k} \theta
Euler's formula
e^{i\pi} + 1 = 0 (Euler's identity)
e^{i\theta} = \cos \theta + i \sin \theta (Euler's formula)
(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta (De Moivre's formula)