How to solve this equation with symmetric polynomials?

Question

Here is what I did

$$x + \sqrt {17 - x^2} + x\times\sqrt{17 - x^2} = 9$$ I can`t undestand how to solve it, any help would be appreciated!

Answer 1

By using your substitution ($y=\sqrt{17-x^2}$), we get

$$ \begin{cases} {x+y+xy=9} \\ {x^2+y^2=17} \end{cases} $$

From the first equation we have,

$$(x+y)^2=(9-xy)^2\;\Rightarrow\ 17+2xy=(xy)^2-18(xy)+81$$ And by solving the quadratic equation in $xy$ we have $xy\in\{16,4\}$. Hence there are two cases, $$ \begin{cases} {xy=16} \\ {x+y=-7} \end{cases} \qquad \text{or} \qquad \begin{cases} {xy=4} \\ {x+y=5} \end{cases} $$

Can you finish it now?