STEP III 1990 question 1 solution

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By expanding and dividing the imaginary part by the real part you get:

Where s = , c = .

Dividing by , you get:

Where .

By expanding the brackets, we see this is equivalent to .

The roots of this equation are given by the roots of the numerator. The roots are .

Since and using the difference of two squares along with the known values for , we get the roots as .

This is because, for example .

So can be factorised into . Examining the constant on both we get .

Setting , we also have:

.

Solution by Dystopia.