how do you rewrite a recursive formula to find its roots

Question

Let $(x_n)$ be the sequence defined by $x_1=2$ and the recursive formula $x_{n+1} = \frac12 + \sqrt{x_n}$.

Rewrite the recursive formula in the form

$$ x_n - x_{n+1} = ax_{n+1}^2 + bx_{n+1} + c$$

and find the roots of the quadratic polynomial on the right hand side.

Answer 1

Hint: Use the recursive formula to get rid of $x_{n+1}$ in the target equation and see if you can collect matching summands, thus finding good suggestions for $a,b,c$.