Find the square root of-7-24i

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

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