I am working on a math puzzle that results in the answer setting up a pair of equations for corresponding sides of similar triangles, then solving the first for y and substituting in the second that gives an equation with a single unknown, like this:
$$x - \frac {x} {\sqrt {16 - x^2}} - \frac {x} {\sqrt {9 - x^2}} = 0$$
Now the trick is to solve for $x$. But it has been pointed out to me that this equation is a quartic. OK, there are lots of places on the 'net that can solve the roots of quartics, no problem, as computers have come a long way. But how do I, when given a polynomial of this type, deduce that it is a cubic or a quartic or even a quintic, and solve for it but not have the coefficients of the general form? Since this equation has no $x^4$th term in it, how do I know that I'm dealing with a quartic? How can I manipulate this equation to get to the general form of $ax^4 + bx^3 + cx^2 +dx + e = 0$, thus having numbers for $a, b, c, d$, and $e$? (My TI-89 using nSOLVE gives the answer as $\pm 2.60328775442$, and $0$ for $x$, thus giving me only 3 solutions, not 4, making me think that it is a cubic)
If anyone would like the complete puzzle to see what I am working on, please ask, I am happy to supply the puzzle!
Thanks for any help!!