Algebra with a complicated function

• Nusc
In summary, the conversation discusses a solution obtained from Mathematica for an equation involving \alpha and \beta as integers. The solution is messy and the person is looking for a way to generalize it concisely and find a general expression for x in the equation. They mention two conditions that must be taken into account, \alpha \geq 0 and 1+2\alpha < 4\beta, and wonder if it is possible to generalize it. They also mention that based on the equality for the first condition, it can be determined that \beta > 0. However, they are unsure if the equation can be generalized at all.

Homework Statement

$$\frac{\pi(1+2\alpha)}{t}=x \&\& ((\alpha \geq 0 \&\& 1+ 2\alpha < 4\beta \&\& \pi \sqrt{-(1+2\alpha)^2+16\beta}=2t)\\ ||((1+2\alpha>0 \&\& 2\alpha < 1 +4 \beta \&\& \beta \geq 0 \&\& \pi \sqrt{(-1+2\alpha -4\beta) (3+2 \alpha +4\beta)}=2t)$$

$$\alpha and \beta$$are 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?

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?

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Based on the equality for condition 1, we also know that $$\beta >0$$ 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.

1. What is a complicated function in algebra?

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