Golden Mean as Root of Quadratic
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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:
- $\hat \phi = 1 - \phi = \dfrac {1 - \sqrt 5} 2$
$\blacksquare$