- #1

idmena

- 14

- 0

## Homework Statement

I want to solve:

$$ x+a(x^2-b)^{1/2}+c=0$$

## Homework Equations

## The Attempt at a Solution

My approach is to make a change of variable x=f(y) to get a true polynomial (integer powers) that I know how to solve, e.g.:

$$y^2+a y +b =0 $$

Then I can switch back from y to x and use each of the solutions in y to get solutions in x. I find that works ok when the power of x inside the root is lower or equal than the power of x outside the root, for example, for the following equations:

$$ x+a(x-b)^{1/2}+c=0 $$

$$ x^2+a(x-b)^{1/2}+c=0 $$

$$ x^2+a(x^2-b)^{1/2}+c=0 $$

the solution is given by making the replacement: ##y=(x^n-b)^{1/2}##, then you have a polynomial in y, e.g.: ##y^2+ay+(b+c)=0## which is straightforward.

If I try the same on this one I get a fractional power of y in the new polynomial, which puts me back in square 1. So far I have not been able to find the right change of variable for this problem.

I am trying to work my way up to:

$$ a x + b x^2 + c x^3 + d (e+fx+(g+hx+jx^2)^{1/2})^2 =0$$

which is the actual equation I need to solve in the problem I am working on.

Any hints?

Thank you!