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

Given that 𝑧 = βˆ’8𝑖, determine the square roots of 𝑧 without first converting to trigonometric form.

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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 𝑀 equal to π‘₯ plus 𝑖𝑦 so that π‘₯ is the real part and 𝑦 is the coefficient of the complex part of our square root. So then, if 𝑀 is the square root of 𝑧, then 𝑀 squared is 𝑧, which is negative eight 𝑖. And since 𝑀 is π‘₯ plus 𝑖𝑦, then π‘₯ plus 𝑖𝑦 squared must be negative eight 𝑖. If we expand our left-hand side, we have π‘₯ squared plus two 𝑖π‘₯𝑦 minus 𝑦 squared is negative eight 𝑖. And comparing real and imaginary parts on the left-hand side, we have π‘₯ squared minus 𝑦 squared is equal to zero and two π‘₯𝑦 is negative eight.

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

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

Our equation one tells us that π‘₯ squared is equal to 𝑦 squared so that if π‘₯ is positive or negative two, then the corresponding 𝑦-values are negative or positive two. Remembering that our square roots were 𝑀 which is equal to π‘₯ plus 𝑖𝑦 and using the values of π‘₯ 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.

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