Fascinating for Centuries

$0 = \phi -1 - \frac{1}{\phi}$ multiplying both sides by $\phi$ you get

$0 = \phi^{2} - \phi - 1$ now use the quadratic formula

$\phi$ { $\frac{1+ \sqrt{5}}{2} , \frac{1 - \sqrt{5}}{2}$ } $\approx$ {1.618 , -0.618}

The positive solution is called the Golden Mean or the Golden Ratio. Ancient mathematicians were very interested in it.