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Calculus and Beyond Homework Help
Solution of "polynomial" with integer and fractional powers
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[QUOTE="idmena, post: 5498913, member: 447078"] Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers. [h2]Homework Statement [/h2] I want to solve: $$ x+a(x^2-b)^{1/2}+c=0$$ [h2]Homework Equations[/h2] [h2]The Attempt at a Solution[/h2] 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! [/QUOTE]
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Solution of "polynomial" with integer and fractional powers
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