Q Find the square root of the complex number 5 -12i - Maths - Complex Numbers and Quadratic Equations

Question

Q Find the square root of the complex number 5
-12i

Ankush Jain · Accepted Answer

Let $\sqrt{5 - 12 i} = x + i y$ â‡’ 5 - 12 i = (x + iy)2 â‡’ 5 - 12 i = x2 - y2 + 2i xy â‡’ 5 - 12 i = (x2 - y2) + 2i xy â‡’ x2 - y2 = 5 (1) and 2xy = -12 Now, $(x^{2} + y^{2})^{2} = (x^{2} - y^{2})^{2} + 4 x^{2} y^{2} \Rightarrow (x^{2} + y^{2})^{2} = (5)^{2} + (- 12)^{2} \Rightarrow (x^{2} + y^{2})^{2} = 25 + 144 = 169 \Rightarrow x^{2} + y^{2} = 13 (2)$ Solving equation (1) and (2), x2 = 9 and y2 = 4 â‡’ x = Â± 3 and y = Â± 2 From equation (1), 2xy is negative, so x and y are of opposite signs âˆ´ x = 3, y = â€“ 2 or x = â€“ 3, y = 2 $∴ \sqrt{5 - 12 i} = \pm (3 - 2 i)$

Q Find the square root of the complex number 5 -12i.