A Problem on Theory Of Equations

Question

Let $f(x) = x^2 + x$, for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y) = f(z) = m + \sqrt{n}$ and $f(1/y) + f(1/z) = 1/10$ . Compute $100m + n$.

Answer 1

Hint. Let $a=m+\sqrt{n}>0$. Note that $y$ and $z$ are the two solutions of the quadratic equation $x^2+x-a=0$. Therefore $y+z=-1$ and $yz=-a$. Hence $$\frac{1}{y}+\frac{1}{z}=\frac{y+z}{yz}=\frac{-1}{-a}=\frac{1}{a}.$$ Moreover $$\frac{1}{y^2}+\frac{1}{z^2}=\frac{y^2+z^2}{y^2z^2}=\frac{(y+z)^2-2yz}{(yz)^2}=\frac{(-1)^2-2(-a)}{(-a)^2}=\frac{1}{a^2}+\frac{2}{a}.$$ Can you take it from here?