# How do you solve using the completing the square method a²-a-1=0?

Mar 21, 2016

$a = \frac{1 \pm \sqrt{5}}{2}$

#### Explanation:

First multiply the whole equation by ${2}^{2} = 4$ to cut down on the fractions involved.

$0 = 4 {a}^{2} - 4 a - 4$

$= 4 {a}^{2} - 4 a + 1 - 5$

$= {\left(2 a - 1\right)}^{2} - {\left(\sqrt{5}\right)}^{2}$

$= \left(\left(2 a - 1\right) - \sqrt{5}\right) \left(\left(2 a - 1\right) + \sqrt{5}\right)$

$= \left(2 a - 1 - \sqrt{5}\right) \left(2 a - 1 + \sqrt{5}\right)$

Hence:

$a = \frac{1 \pm \sqrt{5}}{2}$