How to show given optimization problem has unique optimal solution for each $a \in \mathbb{R}$ without solving it?

Question

Consider the problem \begin{equation} \mbox{min}~~~ x_1^2+2x_2^2+x_1 \\ s.t.~~~ x_1+x_2\le a, \end{equation} where $a\in \mathbb{R}$ is a parameter. I tried to prove it by contradiction by assuming it has two optimal solutions, say, $(x_1,x_2)$ and $(y_1,y_2)$. Then I equated $x_1^2+2x_2^2+x_1=y_1^2+2y_2^2+y_1$ with $x_1+x_2\le a $ and $y_1+y_2\le a$. But from here I am unable to show that $(x_1,x_2)=(y_1,y_2)$. Can anyone please give me a hint or point me in a right direction?