What is the square root of -24 + 70i?

Let the square root of -24 + 70i be a +
ib.

(a + ib)^2 = -24 +
70i

=> a^2 - b^2 + 2abi = -24 +
70i

equating the real and complex
parts

a^2 - b^2 = -24

2ab =
70

=> ab = 35

=>
b = 35/a

substitute in a^2 - b^2 =
-24

=> a^2 - (35/a)^2 =
-24

=> a^4 - 35^2 =
-24a^2

=> a^4 + 24a^2 - 35^2 =
0

let x = a^2

=> x^2 +
24x - 35^2 = 0

=> x^2 + 49x - 25x - 1225 =
0

=> x(x + 49) - 25(x + 49) =
0

=> (x - 25)(x + 49) =
0

=> x = 25 and x =
-49

but x = a^2

a^2 = 25
=> a = 5 and -5

a^2 = -49 gives complex values of a
but a is a real number, so we ignore this root.

a = 5 , b =
35/5 = 7

a = -5 , b = 35/-5 =
-7

The required square root is 5 + 7i and -5
- 7i