Algebra problem

[SumitG]

Now that UIL is over for this year, we can stop stressing over it and do math for fun!

Find all solutions [unparseable or potentially dangerous latex formula] for the equation given that [unparseable or potentially dangerous latex formula].

[unparseable or potentially dangerous latex formula]

[sqrtneg1]

Let [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula]. The equation can then be rewritten as

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Since [unparseable or potentially dangerous latex formula], we proceed by casework.

Case 1: [unparseable or potentially dangerous latex formula]

Then, [unparseable or potentially dangerous latex formula], and since [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula], we find that

[unparseable or potentially dangerous latex formula].

However, this would cause the original expression to be undefined, and hence no solutions in this case.

Case 2: [unparseable or potentially dangerous latex formula]

This implies that [unparseable or potentially dangerous latex formula].The original expression is therfore undefined, and so there are no solutions in this case.

Case 3: [unparseable or potentially dangerous latex formula]

This implies that [unparseable or potentially dangerous latex formula], and so [unparseable or potentially dangerous latex formula] is a solution.

Case 4: [unparseable or potentially dangerous latex formula]

This implies that [unparseable or potentially dangerous latex formula]. Since [unparseable or potentially dangerous latex formula], there are no solutions in this case.

Case 5: [unparseable or potentially dangerous latex formula]

This implies that [unparseable or potentially dangerous latex formula]. Hence, there are no solutions in this case.

Case 6: [unparseable or potentially dangerous latex formula]

This implies that [unparseable or potentially dangerous latex formula]. So, no solutions in this case.

Therefore we can finally conclude that the only solution is [unparseable or potentially dangerous latex formula].

[SumitG]

Yup! Nice job! That's correct. The solution I had was a little different, but in essence the same.
Another one:

Find the number of solutions to the equation [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] are prime and [unparseable or potentially dangerous latex formula].

[stupidityismygam]

[SumitG]

Hmm I didn't think of it that way at first, but I can see what you mean.

[stupidityismygam]

[SumitG]

Hmm...a solution using SFFT:
Multiply everything by [unparseable or potentially dangerous latex formula] to get:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

Since x, y, p, and q are integers, and p and q are prime, the number of solutions is just the number of divisors of [unparseable or potentially dangerous latex formula], which is [unparseable or potentially dangerous latex formula].

[stupidityismygam]

Crap...whoops, i didn't multiply out denominators properly...