Golden ratio
Calculation

Two quantities a {\displaystyle a} and b {\displaystyle b} are in the golden ratio φ {\displaystyle \varphi } if

One method for finding a closed form for φ {\displaystyle \varphi } starts with the left fraction. Simplifying the fraction and substituting the reciprocal b / a = 1 / φ {\displaystyle b/a=1/\varphi },

Therefore,

Multiplying by φ {\displaystyle \varphi } gives

which can be rearranged to

The quadratic formula yields two solutions:

Because φ {\displaystyle \varphi } is a ratio between positive quantities, φ {\displaystyle \varphi } is necessarily the positive root. The negative root is in fact the negative inverse − 1 φ {\displaystyle -{\frac {1}{\varphi }}}, which shares many properties with the golden ratio.
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