If $ x^2-bx+c=0$ has real roots, prove that both roots are greater than $1$, when $c+1>b>2$.
I tried to prove the given inequality by taking roots greater than $1$.
Let $\alpha$, $\beta$ be the roots of the quadratic equation. So $$\alpha+\beta =b$$ $$\alpha\cdot\beta =c$$
Since $\alpha>1$ and $\beta>1$, it can be deduced that, $$\alpha+\beta >2\implies b>2$$ $$\alpha\cdot\beta >1\implies c>1\implies c+1>2$$ to combine these two inequalities I need another link between $c$ and $b$. How to proceed? Thanks.