Golden Mean as Root of Quadratic

Theorem
The golden mean $\phi$ is one of the roots of the quadratic equation:


 * $x^2 - x - 1 = 0$

The other root is $\hat \phi = 1 - \phi$.

Proof
By Solution to Quadratic Equation:

Thus
 * $x = \dfrac {1 + \sqrt 5} 2$

and:
 * $x = \dfrac {1 - \sqrt 5} 2$

The result follows:

By definition of golden mean:
 * $\phi = \dfrac {1 + \sqrt 5} 2$

From Closed Form of One Minus Golden Mean:
 * $\hat \phi = 1 - \phi = \dfrac {1 - \sqrt 5} 2$