The question

Find the best fit to the set of points {(0, 1) (1, 1) (2, 2) (4, 3)} in $\displaystyle R^2$ within the curve $\displaystyle y = ax^2 + bx + c$

My attempt

I used the formula $\displaystyle A^TAx = A^Ty$, and got the following equation:

$\displaystyle \begin{pmatrix}273 & 73 & 21\\73 & 21 & 7\\21 & 7 & 4\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}57\\17\\7\end{pmatrix}$

I tried to use Gaussian elimination, and made a massive mess of the matrix. I was far off a correct solution. Is there a trick to getting an answer without grinding through row operations (or, at least, minimising the work)?

Cheers.