Answer
The required solution is $\left\{ \frac{1-\sqrt{5}}{2},\frac{1+\sqrt{5}}{2} \right\}$
Work Step by Step
Let us consider the given expression:
${{u}^{2}}-u-1=0$
The expression is in the form of the quadratic equation. Therefore, by using the formula $x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$, the expression can be solved.
So, in this question, the value of $a$ is $1$, $b$ is $-1$, and $c$ is $-1$.
Now, the expression can be evaluated as shown below:
$\begin{align}
& x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-\left( -1 \right)\pm \sqrt{{{\left( -1 \right)}^{2}}-4\left( 1 \right)\left( -1 \right)}}{2\left( 1 \right)} \\
& =\frac{1\pm \sqrt{1+4}}{2} \\
& =\frac{1\pm \sqrt{5}}{2}
\end{align}$
Thus, the possible values are $\frac{1-\sqrt{5}}{2}$ and $\frac{1+\sqrt{5}}{2}$.
