Solve a^2=(1a1/sqrt{2})^2+(frac{2sqrt{2}}{2})^2

$$a ^ { 2 } = ( \frac { 1 a 1 } { \sqrt { 2 } } ) ^ { 2 } + ( \frac { 2 \sqrt { 2 } } { 2 } ) ^ { 2 }$$

Solve for a_1 (complex solution)

$$a_{1}=-\sqrt{2\left(a^{2}-2\right)}$$

$$a_{1}=\sqrt{2\left(a^{2}-2\right)}$$

Solve for a

$$a=\frac{\sqrt{2\left(a_{1}^{2}+4\right)}}{2}$$

$$a=-\frac{\sqrt{2\left(a_{1}^{2}+4\right)}}{2}$$

Solve for a_1

$$a_{1}=\sqrt{2\left(a^{2}-2\right)}$$

$$a_{1}=-\sqrt{2\left(a^{2}-2\right)}\text{, }|a|\geq \sqrt{2}$$