Example of finding the square root of a complex number What is the square root of -­‐5 ʹ 12i? The square root of a complex number will not be a real number, and so we can write the following equation: Now to simplify (get rid of the square root) the left side, we can square both sides: Or (if we square the left and rearrange the right side so that the parts that will create the real number are together and the imaginary part is separate) Now we can write a system of equations: tĞĐĂŶ͛ƚƵƐĞůŝŶĞĂƌĐŽŵďŝŶĂƚŝŽŶso we use substitution with the second equation: And now we get a common denominator of b2: Moving everything to the right, I get: 0
or =0 If a fraction equals 0, then the numerator must be 0, therefore we now have: tŚŝĐŚǁĞ͛ƌĞĂďůĞƚo factor as Setting each of those factors equal to 0, we can conclude that b2= 9 or -­‐4 and b = 3 (from the 9; the -­‐4 yields an imaginary solution). Jumping above, we substitute for b in , then a = -­‐2 OR And so our answer is 2 ʹ 3i or -­‐2 + 3i , then a = 2