Algebra with a complicated function

[Nusc]

Homework Statement

<br /> \frac{\pi(1+2\alpha)}{t}=x \&amp;\&amp; ((\alpha \geq 0 \&amp;\&amp; 1+ 2\alpha &lt; 4\beta \&amp;\&amp; \pi \sqrt{-(1+2\alpha)^2+16\beta}=2t)\\ ||((1+2\alpha&gt;0 \&amp;\&amp; 2\alpha &lt; 1 +4 \beta \&amp;\&amp; \beta \geq 0 \&amp;\&amp; \pi \sqrt{(-1+2\alpha -4\beta) (3+2 \alpha +4\beta)}=2t)<br />

<br /> \alpha and \betaare integers.

This is a solution I obtained from Mathematica, it's ugly as you can tell. How can I generalize this concisely?
How can I find a general expression for x from this equation?

Homework Equations

The Attempt at a Solution

If we look at the first condition, \alpha \geq 0 which is nice
\alpha \beta 1+2*alpha < 4*beta
0 1 1 < 4
1 2 3 < 8
2 3 5 < 12This is false when
\alpha \beta 1+2*alpha < 4*beta
0 0 1 < FALSE
1 1 3 < 4
2 2 5 < 8

Any suggestions?

 

[Nusc]

Based on the equality for condition 1, we also know that \beta &gt;0 but how do you generalize the order?

Can this even be generalized to start with? I think not, but how can you tell? There are two condition I have to take into account.