$$\left. \begin{array} { l } { a ^ { 2 } + \frac { 1 } { a ^ { 2 } } = 23 } \\ { a + \frac { 1 } { a } } \end{array} \right.$$
$a=\frac{\sqrt{21}-5}{2}\approx -0.208712153\text{, }b=-5$
$a=\frac{5-\sqrt{21}}{2}\approx 0.208712153\text{, }b=5$
$a=\frac{-\sqrt{21}-5}{2}\approx -4.791287847\text{, }b=-5$
$a=\frac{\sqrt{21}+5}{2}\approx 4.791287847\text{, }b=5$
