Problem - 2094
Let $\alpha$ and $\beta$ be two real roots of the equation $x^2 + x - 4=0$. Find the value of $\alpha^2 - 5\beta + 10$ without computing the value of $\alpha$ and $\beta$.
We have $\alpha + \beta = -1$ and $\alpha\cdot\beta = -4$.
Let $A=(\alpha^2 -5\beta+10)$ and $B=(\beta^2-5\alpha + 10)$, then
$$
\begin{array}{}
A+B=&(\alpha+\beta)^2 -2\alpha\beta-5(\alpha+\beta)+20&=34\\
A -B=&(\alpha-\beta)(\alpha+\beta+5)&=4(\alpha-\beta)
\end{array}
$$
Meanwhile,
$$(\alpha-\beta)^2 = (\alpha+\beta)^2-4\alpha\beta = 17 \implies \alpha-\beta=17$$
Therefore $$A-B=\pm 4\sqrt{17} \implies A=\alpha^2 - 5\beta+10=\boxed{17\pm 2\sqrt{17}}$$