find the square root of 5+12i -4-3i - Maths - Complex Numbers and Quadratic Equations

Question

find the square root of

5+12i
-4-3i

Ankush Jain · Accepted Answer

1) 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 (2) Now, $(x^{2} + y^{2})^{2} = (x^{2} - y^{2})^{2} + 4 x^{2} y^{2} \Rightarrow (x^{2} + y^{2})^{2} = (5)^{2} + (12)^{2} \Rightarrow (x^{2} + y^{2})^{2} = 25 + 144 = 169 \Rightarrow x^{2} + y^{2} = 13 (3)$ Solving equation (1) and (3), x2 = 9 and y2 = 4 â‡’ x = Â± 3 and y = Â± 2 From equation (2), 2xy is positve, so x and y are of the same sign âˆ´ x = 3, y = 2 or x = â€“ 3, y = -2 $∴ \sqrt{5 + 12 i} = \pm (3 + 2 i)$ 2) Similarly, you can find out the square root of -4-3i