### Video Transcript

Given that π§ is negative 28 plus 96π, determine the square roots of π§ without first converting it to trigonometric form.

Since we want to find the square roots of the complex number π§ is negative 28 plus 96π, letβs write our square roots as π§ to the power of half is π€ is π₯ plus ππ¦. This then means that π§ is equal to π€ squared. And so we have π§ which is negative 28 plus 96π is equal to π₯ plus ππ¦ squared. If we expand our right-hand side and equate with π§, comparing real and imaginary parts gives us negative 28 is π₯ squared minus π¦ squared and 96 is equal to two π₯π¦.

Our second equation tells us that π₯π¦ is 96 over two, which is 48. And this tells us that both π₯ and π¦ 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 π§, which is π plus ππ, is the square root of π squared plus π squared, so that the modulus squared is π squared plus π squared. And we also know that the square of the modulus of π€, thatβs our root, is equal to the modulus of our original complex number π§. That is, π₯ squared plus π¦ 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 π¦ squared is equal to 100.

And now we have a system of two equations we can solve for π₯ and π¦. That is, π₯ squared minus π¦ squared is negative 28, and π₯ squared plus π¦ 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 donβt put π₯ squared, which is 36, in equation two, we have 36 plus π¦ squared is equal to 100. That is, π¦ squared is 100 minus 36, which is 64.

And this gives us π¦ is positive or negative eight. So we have π₯ is positive or negative six and π¦ is positive or negative eight. And remember that π₯ and π¦ must have the same signs, so that the square roots of π§ is equal to negative 28 plus 96π must be six plus eight π and negative one times six plus eight π.