College Algebra (11th Edition)

$x=\left\{ \dfrac{\sqrt{5}-3}{2},\dfrac{\sqrt{5}+3}{2} \right\}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $x^2-\sqrt{5}x-1=0 ,$ use the Quadratic Formula. $\bf{\text{Solution Details:}}$ In the equation above, $a= 1 ,$ $b= -\sqrt{5} ,$ and $c= -1 .$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then \begin{array}{l}\require{cancel} x=\dfrac{-(-\sqrt{5})\pm\sqrt{(-\sqrt{5})^2-4(1)(-1)}}{2(1)} \\\\ x=\dfrac{\sqrt{5}\pm\sqrt{5+4}}{2} \\\\ x=\dfrac{\sqrt{5}\pm\sqrt{9}}{2} \\\\ x=\dfrac{\sqrt{5}\pm3}{2} .\end{array} The solutions are $x=\left\{ \dfrac{\sqrt{5}-3}{2},\dfrac{\sqrt{5}+3}{2} \right\} .$