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:

\(\ds x\)

\(=\)

\(\ds \frac {-\paren {-1} \pm \sqrt {\paren {-1}^2 - 4 \times 1 \times \paren {-1} } } {2 \times 1}\)

Solution to Quadratic Equation

\(\ds \)

\(=\)

\(\ds \frac {1 \pm \sqrt 5} 2\)

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:

$\hat \phi = 1 - \phi = \dfrac {1 - \sqrt 5} 2$

$\blacksquare$