We shall now derive a formula for taking the square root of a complex number. Assuming that it, too, is a complex number,

Using the quadratic formula,

For the formula, I am changing every “±” to “+”, as any use of “-” in those locations leads to the same square root, the negative root or an extraneous root.

Formula for the Square Root of a Complex Number:

Or, instead of independently calculating the denominator of the coefficient of *i*, merely replace it with twice the real term of the square root you are finding, as you have already found that real term.

Examples:

In both examples, you could ignore the denominator of the coefficient of *i* until you’ve determined the real term of your answer, and then just insert twice that term for the unwritten denominator (4 in the first example, 6 in the second). I leave it to the reader to confirm these results.

*TO BE CONTINUED*