Solve Algebraic Equation: ax^(n-1) - x + 1 - a = 0

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In summary, the conversation is about a person studying percolation theory and being stuck on an algebraic equation. The person has found one solution, x=1, but is asking for help on simplifying the equation further. Another person doubts that the equation can be simplified and gives an example with a specific set of values. They suggest dividing by (x-1) to obtain an equation with a lower power of x.
  • #1
Wuberdall
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Hi,

I'm currently studying percolation theory and here I'm stuck on the this "simple" algebraic equation
[tex]ax^{n-1} - x +1 - a = 0[/tex]
clearly, the trivial solution x = 1 solves it. But I'm told that
[tex]x = 1 - \frac{2a(n-1)-2}{a(n-2)(n-1)}[/tex]
is another solution. This makes we wonder, if the equation can be put into a simpler form (e.g. quadratic or cubic) — even though I haven't succeed to.

Can any of you help ??

Thaks in advance.
 
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  • #2
I don't think that the statement is true. With a=4, n=4, I get x=1/12 and f(1/12) = -1331/432
 
  • #3
Wuberdall said:
clearly, the trivial solution x = 1 solves it.

Then you can obtain an equation in a lower power of x by dividing the original equation by (x-1). Try that for a particular case like n = 3.
 
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What is an algebraic equation?

An algebraic equation is a mathematical statement that uses symbols and operations to express a relationship between two or more variables. It typically involves solving for an unknown value.

What does "ax^(n-1) - x + 1 - a = 0" mean?

This specific algebraic equation represents a polynomial with a variable, x, raised to a certain power, n-1, and multiplied by a coefficient, a. The rest of the terms involve a constant, 1, and the same coefficient, a. The equation is set equal to 0, indicating that we are solving for the value of x that satisfies the equation.

How do I solve this algebraic equation?

To solve this equation, you can use algebraic manipulation and properties to isolate the variable, x, on one side of the equation. This may involve combining like terms, distributing coefficients, and applying inverse operations. Once you have isolated the variable, you can solve for its value using techniques such as factoring, the quadratic formula, or the rational root theorem.

What are the possible solutions for this equation?

Depending on the values of the coefficient, a, and the exponent, n, there may be multiple solutions for this equation. These solutions can be real or complex numbers. The number of solutions will also depend on the degree of the polynomial, which is equal to the highest exponent in the equation.

Why is solving algebraic equations important in science?

Algebraic equations are used in many scientific fields to represent relationships between variables and make predictions about natural phenomena. They are essential for understanding and solving complex problems in physics, chemistry, engineering, and other disciplines. Knowing how to solve algebraic equations also helps scientists make accurate measurements, analyze data, and develop mathematical models to explain and predict real-world phenomena.

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