Question

Find all complex numbers satisfying the equation $2|z{|}^2+z^2-5+i\sqrt{3}=0$

• A. $\pm (\frac{\sqrt{6}}{2}+\frac{1}{\sqrt{2}}i);\pm (\frac{1}{\sqrt{6}}+\frac{3}{2}i)$
• B. $\pm (\frac{\sqrt{6}}{2}-\frac{1}{\sqrt{2}}i);\pm (\frac{2}{\sqrt{6}}-\frac{3}{2}i)$
• C. $\pm (\frac{\sqrt{6}}{2}-\frac{1}{\sqrt{3}}i);\pm (\frac{1}{\sqrt{6}}-\frac{3}{2}i)$
• D. $\pm (\frac{\sqrt{6}}{2}-\frac{1}{\sqrt{2}}i);\pm (\frac{1}{\sqrt{6}}-\frac{3}{\sqrt{2}}i)$

Answer 1

The correct option is D $\pm (\frac{\sqrt{6}}{2}-\frac{1}{\sqrt{2}}i);\pm (\frac{1}{\sqrt{6}}-\frac{3}{\sqrt{2}}i)$

$Letz=a+ibThus:2(a^2+b^2)+(a^2-b^2+2abi)-5+i\sqrt{3}=0=>3a^2+b^2-5=0and2ab+\sqrt{3}=0=>a=-\frac{\sqrt{3}}{b}Substitutinginthefirst:\frac{9}{4b^2}+b^2=5=>b^2=\frac{9}{2}or\frac{1}{2}=>b=\pm \frac{3}{\sqrt{2}}or\pm \frac{1}{\sqrt{2}}=>a=\mp \frac{1}{\sqrt{6}}or\mp \frac{\sqrt{3}}{\sqrt{2}}=\mp \frac{\sqrt{6}}{2}=>z=\pm (\frac{\sqrt{6}}{2}-\frac{i}{\sqrt{2}})or\pm (\frac{1}{\sqrt{6}}-\frac{3i}{\sqrt{2}})$
Hence, (d) is correct.