Given that 𝑧 is negative 28 plus 96𝑖, determine the square roots of 𝑧 without first converting it to trigonometric form.
Since we want to find the square roots of the complex number 𝑧 is negative 28 plus 96𝑖, let’s write our square roots as 𝑧 to the power of half is 𝑤 is 𝑥 plus 𝑖𝑦. This then means that 𝑧 is equal to 𝑤 squared. And so we have 𝑧 which is negative 28 plus 96𝑖 is equal to 𝑥 plus 𝑖𝑦 squared. If we expand our right-hand side and equate with 𝑧, comparing real and imaginary parts gives us negative 28 is 𝑥 squared minus 𝑦 squared and 96 is equal to two 𝑥𝑦.
Our second equation tells us that 𝑥𝑦 is 96 over two, which is 48. And this tells us that both 𝑥 and 𝑦 must have the same signs. Since positive times positive is positive, negative multiplied by negative is also positive. Now, recalling that the modulus of a complex number 𝑧, which is 𝑎 plus 𝑖𝑏, is the square root of 𝑎 squared plus 𝑏 squared, so that the modulus squared is 𝑎 squared plus 𝑏 squared. And we also know that the square of the modulus of 𝑤, that’s our root, is equal to the modulus of our original complex number 𝑧. That is, 𝑥 squared plus 𝑦 squared is equal to the square root of negative 28 square plus 96 squared. This evaluates to the square root of 10000 which is 100, so that 𝑥 squared plus 𝑦 squared is equal to 100.
And now we have a system of two equations we can solve for 𝑥 and 𝑦. That is, 𝑥 squared minus 𝑦 squared is negative 28, and 𝑥 squared plus 𝑦 squared is 100. If we call these equations one and two and adding them together, we have two 𝑥 squared is equal to negative 28 plus 100. That is, two 𝑥 squared is 72. This gives us 𝑥 squared equal to 36, so that 𝑥 is equal to positive or negative six. If we don’t put 𝑥 squared, which is 36, in equation two, we have 36 plus 𝑦 squared is equal to 100. That is, 𝑦 squared is 100 minus 36, which is 64.
And this gives us 𝑦 is positive or negative eight. So we have 𝑥 is positive or negative six and 𝑦 is positive or negative eight. And remember that 𝑥 and 𝑦 must have the same signs, so that the square roots of 𝑧 is equal to negative 28 plus 96𝑖 must be six plus eight 𝑖 and negative one times six plus eight 𝑖.