My university course handout this formula is written: $Ax^2 + 2Bxy+ Cy^2+ 2Dx + 2Ey + F=0$ but I see more results on Google with this one: $Ax^2 + Bxy+ Cy^2+ Dx + Ey + F=0$.
In my seminar, we always don't care about the 2 but It still confuses me. Each time I see the formula, and the following example: $5x^2 + 8xy + 3y^2+ 2x + 6y + 1$, I want to divide by $2$ to get the "real" $B$, $D$ or $E$. Here, for me, It would be: $A = 5, B = 4, C = 3, D = 1, E = 3, F = 1$.
Thanks
Generally, I see the equation written in the form $$f(\textbf{x})=ax^2+2bxy+cy^2+dx+ey+g$$ This form is useful because it easily allows us to express the function in the form $$f(\textbf{x})=\textbf{x}^TH\textbf{x}+\textbf{x}^T\textbf{h}+g$$ where $H$ is the matrix with $a$ and $c$ along the diagonal and $b$ on the off-diagonals, and $\textbf{h}$ is the vector with entries $d$ and $e$. The matrix $H$ is thus symmetric, and can be diagonalized to rotate this quadratic and express it in the form of an ellipse or hyperbola centered at zero with rotated and translated axes.
So, the purpose of the $2$ is just to help you make the matrix symmetric.