Section 1 Mathematics
Subsection 1.1 The Quadratic Formula
There is a derivation of the quadratic formula that avoids using fractions until the very end. Suppose that we have an equation \(ax^2+bx+c=0\text{,}\) where \(a\neq0\text{.}\) Since \(4a\) is also not equal to \(0\text{,}\) we may multiply each side of the equation by \(4a\) and not lose any information:
\begin{align*}
4a^2x^2+4abx+4ac\amp=0\\
\end{align*}
Subtract \(4ac\) from each side:
\begin{align*}
4a^2x^2+4abx\amp=-4ac\\
\end{align*}
Complete the square on the left side by adding \(b^2\) to each side:
\begin{align*}
4a^2x^2+4abx+b^2\amp=b^2-4ac\\
(2ax+b)^2\amp=b^2-4ac\\
\lvert 2ax+b\rvert\amp=\sqrt{b^2-4ac}\\
2ax+b\amp=\pm\sqrt{b^2-4ac}\\
2ax\amp=-b\pm\sqrt{b^2-4ac}\\
\end{align*}
Since \(a\neq0\text{,}\) we may divide by \(2a\) on each side:
\begin{align*}
x\amp=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\end{align*}
And this is the famous quadratic formula.
Subsection 1.2 Some Amusing Identities
These equations are amusing, although maybe only to mathematicians. Largely taken from math.stackexchange.com.
1
math.stackexchange.com/questions/505367/surprising-identities-equations
\begin{equation*}
\sqrt[\sqrt{2}]{2}=\sqrt{2}^{\sqrt{2}}
\end{equation*}
\begin{equation*}
2592=2^59^2
\end{equation*}
\begin{equation*}
\sqrt{2025}=20+25
\end{equation*}
\begin{equation*}
\sum_{n=1}^{\infty}\frac{1}{n^n}=\int_{0}^1\frac{1}{x^x}\,dx
\end{equation*}
\begin{equation*}
3^3 + 4^4 + 3^3 + 5^5 = 3435
\end{equation*}
\begin{equation*}
\int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx=\int_{-\infty}^{\infty}\frac{\sin^2 x}{x^2}\,dx
\end{equation*}
\begin{equation*}
\log(1+2+3)=\log(1)+\log(2)+\log(3)
\end{equation*}
\begin{equation*}
\int_{0}^{\infty}\frac1{1+x^2}\frac1{1+x^{\pi}}\,dx=\int_{0}^{\infty}\frac1{1+x^2}\frac1{1+x^e}\,dx
\end{equation*}
