Question

If ${(x+iy)}^2=7+24i$, then the value of ${(7+\sqrt{-576})}^{\frac{1}{2}}-{(7-\sqrt{-576})}^{\frac{1}{2}}$ is:

• A. $-3i$
• B. $2i$
• C. $6$
• D. $-6i$

Answer 1

The correct option is D $-6i$

Let $(x+iy{)}^2=7+24i$ ......(i)

$\Rightarrow x^2-y^2+i\left(2xy\right)=7+24i$

$\Rightarrow x^2-y^2=7,xy=12$

$\Rightarrow {\left(\frac{12}{y}\right)}^2-y^2=7$

$\Rightarrow 144-y^4=7y^2$

$\Rightarrow y^4+7y^2-144=0$

$\Rightarrow y^2=\frac{-7\pm 25}{2}=-16$ or $9$

$\Rightarrow y=\pm 3$

Taking conjugate of equation (i), we get

$\overline{(x+iy{)}^2}=\overline{7+24i}$

$\Rightarrow (x-iy{)}^2=7-24i$ ...... $[\because \overline{z^2}={\overline{z}}^2]$

$\Rightarrow (x-iy)=\sqrt{7-24i}$

Now, ${(7+\sqrt{-576})}^{\frac{1}{2}}-{(7-\sqrt{-576})}^{\frac{1}{2}}$

$=\sqrt{7+24i}-\sqrt{7-24i}$

$=x+iy-(x-iy)$

$=2yi=6i$ or $-6i$

Hence, option D is correct.