Golden Ratio Version 2

\[\begin {aligned} \frac{a}{b}&=\frac{a+b}{a}\\ a^2&=ba+b^2\\ a^2-ba&=b^2\\ a^2-2\left(\frac{b}{2}\right)a+\left(\frac{b}{2}\right)^2&=b^2+\left(\frac{b}{2}\right)^2\\ \left(a-\frac{b}{2}\right)^2&=b^2+\frac{b^2}{4}\\ a-\frac{b}{2}&=±\sqrt{\frac{5b^2}{4}}\\ a&=\frac{b}{2}±\frac{\sqrt5b}{2}\\ a&=b\left(\frac{1±\sqrt5}{2}\right) \end {aligned}\]

$\begin{aligned} \phi&\overset{\text{def}}{=}\frac{a}{b}\\ &=\frac{b\left(\frac{1±\sqrt5}{2}\right)}{b}\\ &=\boxed{\frac{1±\sqrt5}{2}} \end{aligned}$

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$