Solving an Equation for y in terms of x How is it possible to solve the equation:
$$ x = \frac{2-y}{y} \sqrt{1-y^2} $$
for $ y $ in terms of $ x $?
I know that it is possible to do it, since WolframAlpha gives a complicated answer but entirely composed of elementary functions. However, I am stumped on how to even start.
Thanks in advance!

The typical method would be to square both sides (possibly picking up extraneous solutions) and then arrange the terms into a polynomial equation in $y$:
$$x = \frac{2-y}{y} \sqrt{1-y^2}$$
$$x^2 = \frac{(2-y)^2}{y^2} (1-y^2)$$
$$y^2 x^2 = (4-4y+y^2)(1-y^2) = (4-4y+y^2) - 4y^2 + 4y^3 - y^4$$
Which yields:
$$y^4 - 4y^3 + (x^2 + 3)y^2 + 4y - 4 = 0$$
If this were simply a quadratic in $y$, you could use the quadratic equation. As it stands this is a fourth order polynomial. There is a procedure to find the roots of such a polynomial, and you can find it through the wiki page on Quartic Polynomials. It's a bit of a mess though.
