If you want $a_0$ to be $0$, then the curve you are trying to fit has equation $y = a_1 x + a_2 x^2$. If you derive the normal equations starting from this equation, you get

$a_1 \sum x_i^2 + a_2 \sum x_i^3 = \sum x_i y_i\\
a_1 \sum x_i^3 + a_2 \sum x_i^3 = \sum x_i^2 y_i$

Or

$\left[\begin{array}{cc|c}
\sum x_i^2 & \sum x_i^3 & \sum x_i y_i\\
\sum x_i^3 & \sum x_i^4 & \sum x_i^2 y_i
\end{array}\right]$