Algebra with a complicated function

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.
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Homework Statement



[tex]
\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)
[/tex]

[tex]
\alpha and \beta[/tex]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?

Homework Equations


The Attempt at a Solution



If we look at the first condition, [tex] \alpha \geq 0 [/tex] 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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  • #2
Based on the equality for condition 1, we also know that [tex]\beta >0[/tex] 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?

A complicated function in algebra refers to a mathematical expression or equation that involves multiple variables, exponents, and/or operations such as logarithms, trigonometric functions, or square roots. These functions are often more complex and require advanced algebraic skills to solve.

2. How do you simplify a complicated function?

Simplifying a complicated function involves using algebraic techniques such as factoring, distributing, and combining like terms. The goal is to rewrite the function in a more concise form without changing its underlying meaning. This can help make the function easier to understand and manipulate.

3. Can you graph a complicated function?

Yes, complicated functions can be graphed using a graphing calculator or by hand. However, the graph may appear more complex and have more features compared to simpler functions. It is important to understand the behavior of the function, such as its domain and range, before attempting to graph it.

4. What are some real-life applications of complicated functions?

Complicated functions are commonly used in various fields such as physics, engineering, economics, and computer science. Examples include modeling the trajectory of a projectile, analyzing the growth of a population, and predicting stock market trends.

5. How can I improve my skills in solving complicated functions?

The best way to improve your skills in solving complicated functions is through practice and understanding the fundamental concepts of algebra. It is also helpful to seek out additional resources such as textbooks, online tutorials, and working with a tutor to reinforce your understanding.

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