Solve M=-2+sqrt{a^2/b^2+b^2/a^2+2}

$$M = - 2 + \sqrt { \frac { a ^ { 2 } } { b ^ { 2 } } + \frac { b ^ { 2 } } { a ^ { 2 } } + 2 }$$

Solve for M (complex solution)

$$M=-2+\sqrt{\left(\frac{a^{2}+b^{2}}{ab}\right)^{2}}$$

$$b\neq 0\text{ and }a\neq 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{ib\sqrt{2\left(\sqrt{M\left(M+4\right)\left(M+2\right)^{2}}-M^{2}-4M-2\right)}}{2}\text{; }a=-\frac{ib\sqrt{2\left(\sqrt{M\left(M+4\right)\left(M+2\right)^{2}}-M^{2}-4M-2\right)}}{2}\text{; }a=-\frac{ib\sqrt{2\left(-\sqrt{M\left(M+4\right)\left(M+2\right)^{2}}-M^{2}-4M-2\right)}}{2}\text{; }a=\frac{ib\sqrt{2\left(-\sqrt{M\left(M+4\right)\left(M+2\right)^{2}}-M^{2}-4M-2\right)}}{2}\text{, }&b\neq 0\text{ and }arg(M+2)<\pi \\a=ib\text{; }a=-ib\text{, }&b\neq 0\text{ and }M=-2\end{matrix}\right.$$

Solve for M

$$M=\frac{-2|a||b|+a^{2}+b^{2}}{|a||b|}$$

$$b\neq 0\text{ and }a\neq 0$$

Solve for a

$$\left\{\begin{matrix}a=\frac{b\sqrt{2\left(\sqrt{M+4}M^{\frac{3}{2}}+M^{2}+2\sqrt{M\left(M+4\right)}+4M+2\right)}}{2}\text{, }&M\geq 0\text{ and }2M\sqrt{M\left(M+4\right)}+2M^{2}+4\sqrt{M\left(M+4\right)}+8M\geq -4\text{ and }b\neq 0\text{ and }M+2\geq 0\\a=-\frac{b\sqrt{2\left(\sqrt{M+4}M^{\frac{3}{2}}+M^{2}+2\sqrt{M\left(M+4\right)}+4M+2\right)}}{2}\text{, }&M\geq 0\text{ and }2M\sqrt{M\left(M+4\right)}+2M^{2}+4\sqrt{M\left(M+4\right)}+8M\geq -4\text{ and }b\neq 0\\a=\frac{b\sqrt{2\left(-\sqrt{M+4}M^{\frac{3}{2}}+M^{2}-2\sqrt{M\left(M+4\right)}+4M+2\right)}}{2}\text{, }&M\geq 0\text{ and }-2M\sqrt{M\left(M+4\right)}+2M^{2}-4\sqrt{M\left(M+4\right)}+8M\geq -4\text{ and }b\neq 0\text{ and }M+2\geq 0\\a=-\frac{b\sqrt{2\left(-\sqrt{M+4}M^{\frac{3}{2}}+M^{2}-2\sqrt{M\left(M+4\right)}+4M+2\right)}}{2}\text{, }&M\geq 0\text{ and }-2M\sqrt{M\left(M+4\right)}+2M^{2}-4\sqrt{M\left(M+4\right)}+8M\geq -4\text{ and }b\neq 0\end{matrix}\right.$$