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

Given that π§ = β28 + 96π, determine the square roots of π§ without first converting it to trigonometric form.

02:29

### 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, letβs write our square roots as π§ to the power of half is π€ is π₯ plus ππ¦. 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 ππ¦ 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 ππ, 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€, thatβs 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 donβt 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.