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

Given that 𝑧 = βˆ’28 + 96𝑖, determine the square roots of 𝑧 without first converting it to trigonometric form.

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Video Transcript

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 𝑀 is π‘₯ plus 𝑖𝑦. This then means that 𝑧 is equal to 𝑀 squared. And so we have 𝑧 which is negative 28 plus 96𝑖 is equal to π‘₯ plus 𝑖𝑦 squared. If we expand our right-hand side and equate with 𝑧, comparing real and imaginary parts gives us negative 28 is π‘₯ squared minus 𝑦 squared and 96 is equal to two π‘₯𝑦.

Our second equation tells us that π‘₯𝑦 is 96 over two, which is 48. And this tells us that both π‘₯ 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 π‘Ž plus 𝑖𝑏, is the square root of π‘Ž squared plus 𝑏 squared, so that the modulus squared is π‘Ž squared plus 𝑏 squared. And we also know that the square of the modulus of 𝑀, that’s our root, is equal to the modulus of our original complex number 𝑧. That is, π‘₯ 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 π‘₯ squared plus 𝑦 squared is equal to 100.

And now we have a system of two equations we can solve for π‘₯ and 𝑦. That is, π‘₯ squared minus 𝑦 squared is negative 28, and π‘₯ squared plus 𝑦 squared is 100. If we call these equations one and two and adding them together, we have two π‘₯ squared is equal to negative 28 plus 100. That is, two π‘₯ squared is 72. This gives us π‘₯ squared equal to 36, so that π‘₯ is equal to positive or negative six. If we don’t put π‘₯ 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 π‘₯ is positive or negative six and 𝑦 is positive or negative eight. And remember that π‘₯ 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.

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