What is the square root of (-7 + 24i)?

Let the required square root of -7 + 24i be x +
yi

x + yi = sqrt ( -7 +
24i)

square both the sides

x^2
- y^2 + 2xyi = -7 + 24i

equate the real and complex
coefficients

x^2 - y^2 =
-7

2xy = 24

=> xy =
12

=> x =
12/y

Substitute in x^2 - y^2 =
-7

=> 12^2/y^2 - y^2 =
-7

=> 144 - y^4 + 7y^2 =
0

=> y^4 - 7y^2 - 144 =
0

let u = y^2

=> u^2 -
7u - 144 = 0

=> u^2 - 16u + 9u - 144 =
0

=> u(u - 16) + 9(u - 16) =
0

=> (u - 16)(u + 9) =
0

=> u = 16 and u =
-9

y = sqrt u is a real number so we take only u =
16.

y^2 = 16 , y = 4 and y =
-4

x = 12/y = 3 and
-3

The required value of sqrt( - 7 + 24i) =
-3 - 4i and 3 + 4i