quadratic equations with eg a term 7/x

[baffledgrandad]

say x^2 + 7/x + 3/x^2 + 9x = 123

I cannot even find the google question that deals with this type of equation. If I multiply by x^2
I get x^4 + 7x +3 +9x^3 = 123 x^3

help please

 

[Romsek]

Well you did the right thing, but you have a typo. It's ... = 123x^2

There don't appear to be any nice solutions. Do you need to solve for them?
Wolfram Alpha will give you numeric solutions.

Dump

Solve[x^4 + 7x +3 +9x^3 = 123 x^2 ,x]

onto their command line

 

[Mr. Bland]

As has been pointed out, one can produce a polynomial by multiplying both sides by [MATH]x^2[/MATH]:

[MATH]x^2 + \frac{7}{x} + \frac{3}{x^2} + 9x = 123[/MATH]​

​

[MATH]x^4 + 7x + 3 + 9x^3 = 123x^2[/MATH]​

​

[MATH]x^4 + 9x^3 - 123x^2 + 7x + 3 = 0[/MATH]​

This is a quartic function (not to be confused with quadratic function), so solving for [MATH]x[/MATH] requires some footwork. While there is a general solution for this, it's staggeringly complicated:

​

I'm personally not sure about what is taught in academic settings regarding the solving of higher-order polynomials. Wolfram|Alpha can show you the steps it takes for any given input, but that's a paid service.

Incidentally, it has been proven that for any polynomial of degree 5 or higher, there is no general formula for finding all of the roots if the coefficients are not restricted.

 

[Deleted member 4993]

If I were asked to find "roots" of this quartic:

I would first approximate those through graphical method.​

​

After realizing that there is no rational root, we can apply Newton-Raphson to approximate the roots (better accuracy than graphical method). The graphical solutions will provide efficient starting points for N-R method.​

 

[JeffM]

If I were asked to find "roots" of this quartic:

I would first approximate those through graphical method.​

​

After realizing that there is no rational root, we can apply Newton-Raphson to approximate the roots (better accuracy than graphical method). The graphical solutions will provide efficient starting points for N-R method.​

I wanted to give this post a double like.

 

[Deleted member 4993]

I wanted to give this post a double like.

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