Question

Solve: $x^2-(3\sqrt{2}-2i)x-\sqrt{2}i=0$

Answer 1

Given,

$x^2-(3\sqrt{2}-2i)x-\sqrt{2}i=0$

${(a+bi)}^2-(3\sqrt{2}-2i)(a+bi)-\sqrt{2}i=0$

$(a^2-b^2-3\sqrt{2}a-2b)+i(-\sqrt{2}+2a+2ab-3\sqrt{2}b)=0+0i$

$\left[\begin{array}{c}{a}^2-b^2-3\sqrt{2}a-2b=0\\ -\sqrt{2}+2a+2ab-3\sqrt{2}b=0\end{array}\right]$

solving the above 2 equations, we get,

$\left(\begin{array}{cc}a=\frac{3\sqrt{2}-4}{2},& b=-\frac{-\sqrt{2}+2}{2}\\ a=\frac{3\sqrt{2}+4}{2},& b=-\frac{2+\sqrt{2}}{2}\end{array}\right)$

$x=\frac{3\sqrt{2}-4}{2}-\frac{-\sqrt{2}+2}{2}i,\frac{3\sqrt{2}+4}{2}-\frac{2+\sqrt{2}}{2}i$