Question

Find the square root of the following complex number:

−5 + 12i

Answer 1

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) > 0\]

\[\sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} - i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right] , \text { if Im }(z) < 0\]

\[ z = - 5 + 12i , Re\left( z \right) = - 5 \text { and } \left| z \right| = \sqrt{25 + 144} = 13\]

\[ \text { Here, Im }(z) > 0\]

\[ \therefore \sqrt{z} = \pm \left[ \sqrt{\frac{\left| z \right| + Re\left( z \right)}{2}} + i\sqrt{\frac{\left| z \right| - Re\left( z \right)}{2}} \right]\]

\[ = \pm \left[ \sqrt{\frac{13 - 5}{2}} + i\sqrt{\frac{13 + 5}{2}} \right]\]

\[ = \pm \left[ \sqrt{4} + i\sqrt{9} \right]\]

\[ = \pm \left( 2 + 3i \right)\]