Answer
$\left\{ \dfrac{9-5\sqrt{5}}{22},\dfrac{9+5\sqrt{5}}{22} \right\}$
Work Step by Step
The standard form of the given equation, $
11n^2-9n=1
,$ is
\begin{array}{l}\require{cancel}
11n^2-9n-1=0
.\end{array}
Using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solutions of the quadratic equation, $
11n^2-9n-1=0
,$ are
\begin{array}{l}\require{cancel}
\dfrac{-(-9)\pm\sqrt{(-9)^2-4(11)(-1)}}{2(11)}
\\\\=
\dfrac{9\pm\sqrt{81+44}}{22}
\\\\=
\dfrac{9\pm\sqrt{125}}{22}
\\\\=
\dfrac{9\pm\sqrt{25\cdot5}}{22}
\\\\=
\dfrac{9\pm\sqrt{(5)^2\cdot5}}{22}
\\\\=
\dfrac{9\pm5\sqrt{5}}{22}
.\end{array}
Hence, the solutions are $
\left\{ \dfrac{9-5\sqrt{5}}{22},\dfrac{9+5\sqrt{5}}{22} \right\}
.$
