Multiply both sides by \(ab+1\).
\[x(ab+1)={a}^{2}+{b}^{2}\]
Expand.
\[xab+x={a}^{2}+{b}^{2}\]
Move all terms to one side.
\[xab+x-{a}^{2}-{b}^{2}=0\]
Use the Quadratic Formula.
In general, given \(a{x}^{2}+bx+c=0\), there exists two solutions where:
\[x=\frac{-b+\sqrt{{b}^{2}-4ac}}{2a},\frac{-b-\sqrt{{b}^{2}-4ac}}{2a}\]
In this case, \(a=-1\), \(b=xb\) and \(c=x-{b}^{2}\).
\[{a}^{}=\frac{-xb+\sqrt{{(xb)}^{2}-4\times -(x-{b}^{2})}}{-2},\frac{-xb-\sqrt{{(xb)}^{2}-4\times -(x-{b}^{2})}}{-2}\]
Simplify.
\[a=\frac{-xb+\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{-2},\frac{-xb-\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{-2}\]
\[a=\frac{-xb+\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{-2},\frac{-xb-\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{-2}\]
Simplify solutions.
\[a=-\frac{-xb+\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{2},-\frac{-xb-\sqrt{{x}^{2}{b}^{2}+4(x-{b}^{2})}}{2}\]
a=-(-x*b+sqrt(x^2*b^2+4*(x-b^2)))/2,-(-x*b-sqrt(x^2*b^2+4*(x-b^2)))/2
