Question Video: Finding the Square Roots of Complex Numbers in Algebraic Form Mathematics

Video Transcript

Given that 𝑧 is negative eight 𝑖, determine the square roots of 𝑧 without first converting to trigonometric form.

Since we want to find the square root of the complex number 𝑧 is negative eight 𝑖, let’s give our square root some form and call them 𝑤 equal to 𝑥 plus 𝑖𝑦 so that 𝑥 is the real part and 𝑦 is the coefficient of the complex part of our square root. So then, if 𝑤 is the square root of 𝑧, then 𝑤 squared is 𝑧, which is negative eight 𝑖. And since 𝑤 is 𝑥 plus 𝑖𝑦, then 𝑥 plus 𝑖𝑦 squared must be negative eight 𝑖. If we expand our left-hand side, we have 𝑥 squared plus two 𝑖𝑥𝑦 minus 𝑦 squared is negative eight 𝑖. And comparing real and imaginary parts on the left-hand side, we have 𝑥 squared minus 𝑦 squared is equal to zero and two 𝑥𝑦 is negative eight.

Our second equation implies 𝑥𝑦 is negative eight over two; that is negative four. And this tells us, for one thing, that the signs of 𝑥 and 𝑦 are not the same. So, if 𝑥 is positive, 𝑦 must be negative and vice versa. Now, recalling that for a complex number 𝑧, which is 𝑎 plus 𝑖𝑏, the modulus of 𝑧 is the square root of 𝑎 squared plus 𝑏 squared. This then means that the modulus of 𝑧 squared is 𝑎 squared plus 𝑏 squared. And applying this to our square root 𝑤, where 𝑤 is 𝑥 plus 𝑖𝑦, we have the modulus of 𝑤 squared is 𝑥 squared plus 𝑦 squared. And this must be equal to the modulus of our original complex number 𝑧 is negative eight 𝑖. That is the square root of zero squared plus negative eight squared, which is eight, that is, that 𝑥 squared plus 𝑦 squared is equal to eight.

And we now have a set of two equations we can solve for 𝑥 and 𝑦. That’s 𝑥 squared minus 𝑦 squared is zero and 𝑥 squared plus 𝑦 squared is eight. Now, just tidying up and making some room, if we call our first equation equation one, that’s 𝑥 squared minus 𝑦 squared is zero. And equation two is 𝑥 squared plus 𝑦 squared is eight. Adding equations one and two gives us two 𝑥 squared is equal to eight. That is, 𝑥 squared is equal to four so that 𝑥 is equal to positive or negative two.

Our equation one tells us that 𝑥 squared is equal to 𝑦 squared so that if 𝑥 is positive or negative two, then the corresponding 𝑦-values are negative or positive two. Remembering that our square roots were 𝑤 which is equal to 𝑥 plus 𝑖𝑦 and using the values of 𝑥 and 𝑦 that we just found, then the square roots of 𝑧 is negative eight 𝑖 are two minus two 𝑖 and negative two plus two 𝑖.