Solve an Equation

[KQL]

I am new to this site. I need help solving the following equation. It is part of a science paper I am trying to write.

X=variable , K and B are constants. K=X+X/√(B^2-X^2 ) , Solve for X. have tried for hours getting nowhere fast. I would appreciate any help

Thanks KQL

 

[Deleted member 4993]

I am new to this site. I need help solving the following equation. It is part of a science paper I am trying to write.

X=variable , K and B are constants. K=X+X/√(B^2-X^2 ) , Solve for X. have tried for hours getting nowhere fast. I would appreciate any help

Thanks KQL

.
First eliminate the "fraction"

K = X + X/√(B^2-X^2 )

K * (√(B^2-X^2 )) = X * (√(B^2- X^2 ) + X

Now continue.....

 

[Otis]

… K = X + X / √(B^2-X^2) , Solve for X. have tried for hours …

Hello KQL. Variable X appears both inside and outside the square-root expression. Therefore, you'll need to eventually square each side. Sometimes, we need to square each side of an equation twice, while solving. This is one of those cases. Yet, squaring twice results in a 4th-degree polynomial. Calculating roots of such polynomials by hand is very tedious.

May I suggest using software? You can see the four solutions here.

Those solutions appear unwieldy because they account for every possible value of K and B. If you know values for K or B, the results are simpler.

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[KQL]

Thank you very much Otis. Your reply is very helpful. I kept reformulating the equation and getting a 4th degree polynomial and wasn't sure if this was inevitable or I was missing something. That the solutions by hand would be tedious and appear unwieldy indeed, you are a master of understatement Sir. I assume the solutions are a positive and negative real number and a positive and negative imaginary number, correct?
K is an experimentally derived positive number. B is a true positive constant , what difference would this make. The Computer analysis of the equation is quite impressive . How would you enter the facts about B and K into the input to see the simpler solutions ? In any case although difficult , if I have my number K from experiment then a solution is able to be computed correct ? This allows me to write about this solution in the paper with confidence. Thanks again

KQL

 

[Otis]

… I assume the solutions are a positive and negative real number and a positive and negative imaginary number, correct? …

Hi KQL. The types of solutions depend upon the values of B and K.

… How would you enter the facts [B>0 and K>0] to see the simpler solutions?

I'm not sure that would simplify much. Unfortunately, I don't know the command/syntax for doing that at WolframAlpha. The Computer Algebra System (CAS) used there is based on Mathematica. I've never had much success, while trying to input Mathematica commands I've found online, or when searching WolframAlpha's site for specific commands. I hope someone else here posts specific instructions.

… if I have my number K from experiment then a solution is able to be computed correct? …

If you replace K with a Real number, then the solutions will be symbolic (in terms of B).

To obtain numerical solutions, you'd need to supply values for both K and B. You can experiment, by replacing symbols K and B with numbers. For example, let's see what happens when K=4 and B=1.5 over the x-interval [0,∞].

solve 4=x+x/sqrt(1.5^2-x^2) for x in [0, infinity]

WolframAlpha reports:

x ≈ 1.40002

Well, that's true, but it's not the complete solution. I suspect the CAS is using a root-finding process, and it stops after the first result.

Extra work is needed (for me, at least, because I don't know their language). With those values of K and B, the polynomial is:

plot 65x^4-272.2500000x^2+284.7656250



It appears there are four Real solutions.

My rusty copy of MVR5 (Maple V Release 5, which I also know how to use for only basic stuff) reports:

x ≈ -1.470871014
x ≈ -1.423024947
x ≈ 1.423024947
x ≈ 1.470871014

Here's a zoomed-in graph of the vicinity around the positive results.



If you're interested in getting more details out of WolframAlpha, I'd suggest joining their users' group and posting questions about what to enter. Alternative sites for posting questions are quora.com and math.stackexchange.com. There might be additional replies posted in this thread, too.

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[KQL]

Thanks again Otis. I feel I can proceed confidently at this point I will explore WolframAlpha for future projects.

KQL