Find the square root of-7-24i

Let the square root of 7 + 24i = a +bi where a and b are real numbers. We know that i^2 = -1 Now (a+bi)^2 = 7 + 24i => a^2 -b^2 + 2(a)(b)i =7 + 24i Comparing both sides we get two equations a^2 - b^2 =7 And 2ab = 24 => ab = 12 => b =12/a Now putting the value of b in equation 1 we get a^2 - (12/a)^2 = 7 => a^2 -(144/a^2) = 7 => a^4 -144 = 7a^2 => a^4 - 7 a^2 -144 = 0 Solving above quadratic equation we get a^2 = (7+√(576+49))/2 or a^2 = (7-√(576+49))/2 => a^2 = (7+25)/2 or a^2= (7–25)/2 => a^2 = 16 or a^2 = -9 Since we have assumed a as real a^2 can't be negative Therefore a^2 = 16 => a = 4 or a = -4 Putting the value of a in ab = 12 we get b = 3 or b = -3 Therefore our required solution is 4 + 3i or -4 -3i