Mathematics Formulas
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\[
\begin{array}{c}
x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\end{array}
\]
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\[
\begin{aligned}
A &= P(1 + ni) \\
A &= P(1 - ni) \\
A &= P(1 - i)^{n} \\
A &= P(1 + i)^{n}
\end{aligned}
\]
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\[
\begin{aligned}
T_{n} &= a + (n-1)d \\
S_{n} &= \dfrac{n}{2}\left[2a + (n-1)d\right] \\
T_{n} &= ar^{n-1} \\
S_{n} &= \dfrac{a\left(r^{n} - 1\right)}{r - 1} \, ; \, r \ne 1 \\
S_{\infty} &= \dfrac{a}{1-r} \, ; \, -1 \lt r \lt 1
\end{aligned}
\]
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\[
\begin{aligned}
F &= \dfrac{x\left[(1+i)^{n} - 1\right]}{i} \\
P &= \dfrac{x\left[1 - (1+i)^{-n}\right]}{i}
\end{aligned}
\]
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\[
\begin{array}{c}
f^{\prime}(x) = \lim_\limits{h \to 0} \dfrac{f(x + h) - f(x)}{h}
\end{array}
\]
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\[
\begin{array}{c}
d = \sqrt{\left(x_{2} - x_{1}\right)^{2} + \left(y_{2} - y_{1}\right)^{2}} \\
M \left(\dfrac{x_{1} + x_{2}}{2} \, ; \, \dfrac{y_{1} + y_{2}}{2}\right) \\
y = mx + c \\
y - y_{1} = m(x - x_{1}) \\
m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \\
m = \tan\theta \\
(x - a)^{2} + (y- b)^{2} = r^{2}
\end{array}
\]
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\[
\begin{array}{c}
\text{In }\Delta{ABC}: \\
\dfrac{a}{\sin{A}} = \dfrac{b}{\sin{B}} = \dfrac{c}{\sin{C}} \\
a^{2} = b^{2} + c^{2} - 2bc{\cdot}\cos{A} \\
\text{area }\Delta{ABC} = \dfrac{1}{2}ab{\cdot}\sin{C}
\end{array}
\]
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\[
\begin{aligned}
\sin\left(\alpha + \beta\right) &= \sin{\alpha}{\cdot}\cos{\beta} + \cos{\alpha}{\cdot}\sin{\beta} \\
\sin\left(\alpha - \beta\right) &= \sin{\alpha}{\cdot}\cos{\beta} - \cos{\alpha}{\cdot}\sin{\beta} \\
\cos\left(\alpha + \beta\right) &= \cos{\alpha}{\cdot}\cos{\beta} - \sin{\alpha}{\cdot}\sin{\beta} \\
\cos\left(\alpha - \beta\right) &= \cos{\alpha}{\cdot}\cos{\beta} + \sin{\alpha}{\cdot}\sin{\beta} \\[0.2cm]
\cos{2\alpha} &= \left\{
\begin{aligned}
& \cos^{2}\alpha - \sin^{2}\alpha \\
& 1 - 2\sin^{2}\alpha \\
& 2\cos^{2}\alpha - 1
\end{aligned}
\right. \\
\sin{2\alpha} &= 2\sin\alpha{\cdot}\cos\alpha
\end{aligned}
\]
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\[
\begin{aligned}
\bar{x} &= \dfrac{\sum{x}}{n} \\
\sigma^{2} &= \dfrac{{\sum^{n}_{i=1}{\left(x_{i} - \bar{x}\right)^{2}}}}{n} \\
P(A) &= \dfrac{n(A)}{n(S)} \\
P(A \text{ or } B) &= P(A) + P(B) - P(A \text{ and } B) \\
\hat{y} &= a + bx \\
b &= \dfrac{\sum{\left(x - \bar{x}\right)\left(y - \bar{y}\right)}}{\sum{\left(x - \bar{x}\right)^{2}}}
\end{aligned}
\]
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