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A Spesh question from me

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Glockmeister:
From my assignment

If , where , show that the number is real only if

Collin Li:
If is real, then and .









For to be real,





Where (as given):







as required.

enwiabe:
That is a really neat question. Nice working, coblin.

Collin Li:

--- Quote from: enwiabe on February 04, 2008, 10:11:32 pm ---That is a really neat question. Nice working, coblin.

--- End quote ---

Thanks. I love complex number questions. In 2006 I had the feeling that I had a comparative advantage in this topic compared to the state. They have a lot of meaning, but it's largely ignored in the VCE syllabus.

Neobeo:
Alternative solution,

We can rearrange the equation into a quadratic for z, and get
Applying the quadratic formula gives

Since we are given that , must have an imaginary part. Also since is real, it follows that must be imaginary. (Incidentally we also prove that ).

The rest is just basic substitution:

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