Revisions to Solution of a sixth degree polynomial

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edited Nov 23, 2023 at 8:49

101
6

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following code

DefiningExpression = Simplify[r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2)]; sol = Solve[DefiningExpression == 0, r]; CloseBound = 2 M (1 + Cos[2/3 ArcCos[-a/M]]); FarBound = 2 M (1 + Cos[2/3 ArcCos[a/M]]);   Plot[{Lookup[sol[[1]], r], Lookup[sol[[2]], r], Lookup[sol[[3]], r], Lookup[sol[[4]], r]}, {x, 0, 2}]

I want to solve the DefiningExpression for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

When plotting this, we get that the solutions between these bounds are spread out between the different solutions of sol:

Here $a=0.7$, $M=1$.

Also the constraints on $a$ and $M$ are that $r>M>a>0$.

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following code

DefiningExpression = Simplify[r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2)]; Solve[DefiningExpression == 0, r]; CloseBound = 2 M (1 + Cos[2/3 ArcCos[-a/M]]); FarBound = 2 M (1 + Cos[2/3 ArcCos[a/M]]);

I want to solve the DefiningExpression for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

Also the constraints on $a$ and $M$ are that $r>M>a>0$.

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following code

DefiningExpression = Simplify[r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2)]; sol = Solve[DefiningExpression == 0, r]; CloseBound = 2 M (1 + Cos[2/3 ArcCos[-a/M]]); FarBound = 2 M (1 + Cos[2/3 ArcCos[a/M]]);   Plot[{Lookup[sol[[1]], r], Lookup[sol[[2]], r], Lookup[sol[[3]], r], Lookup[sol[[4]], r]}, {x, 0, 2}]

I want to solve the DefiningExpression for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

When plotting this, we get that the solutions between these bounds are spread out between the different solutions of sol:

Here $a=0.7$, $M=1$.

Also the constraints on $a$ and $M$ are that $r>M>a>0$.

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?

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edited Nov 23, 2023 at 8:37

Geigercounter

101
6

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following equation $$r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2) = 0.$$code

DefiningExpression = Simplify[r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2)]; Solve[DefiningExpression == 0, r]; CloseBound = 2 M (1 + Cos[2/3 ArcCos[-a/M]]); FarBound = 2 M (1 + Cos[2/3 ArcCos[a/M]]);

I want to solve thisthe DefiningExpression for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

Also the constraints on $a$ and $M$ are that $r>M>a>0$.

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following equation $$r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2) = 0.$$ I want to solve this for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?

I want to get the solution to a 6th degree polynomial in a certain range.

I have the following code

DefiningExpression = Simplify[r^4 (-3 M + r)^2 - a^4 (-1 + x) (M + r)^2 + 2 a^2 r^2 (3 (-1 + x) M^2 - 2 M r - (-1 + x) r^2)]; Solve[DefiningExpression == 0, r]; CloseBound = 2 M (1 + Cos[2/3 ArcCos[-a/M]]); FarBound = 2 M (1 + Cos[2/3 ArcCos[a/M]]);

I want to solve the DefiningExpression for $r$ in terms of $x$. Unfortunately this is a no-so-friendly 6th degree polynomial, so we won't find any analytic solutions. The additional requirement is that only the following values of $r$ are of interest: $$r \in [2 M (1 + \cos[2/3 \arccos[-a/M]]), 2 M (1 + \cos[2/3 \arccos[a/M]])].$$

Also the constraints on $a$ and $M$ are that $r>M>a>0$.

How can I get a solution of $r$ in terms of $x$ where $r$ lies within this range?