let us suppose we have following quadratic forms and its corresponding partial derivative with respect of $x_1$
i have seen following intrpretation
i really did not understand one fact : if $\beta_{2}$ is negative , then line is decreasing, but it says that y is increasing, what is the reason of this ? also if $\beta_{2}$ is negative , and we consider first three part of original equation, we will get parabola with negative first coefficients, therefore arrows of parabola will be down, so could you explain in shortly what is the idea behind of given assumption?
If $\beta_2 < 0$ you have "frown" parabola with respect to $x_1$, namely an increase in one unit in $x_1$ will increase $y$ by $\beta_1 + 2\beta_2x_1$ units until its maximum, while the the rate of increase will gradually slow dawn until the curve reaches its maximum point at $-\frac{\beta_1}{2\beta_2}$. For $x_1 > -\frac{\beta_1}{2\beta_2}$ an increase in $x_1$ will decrease $y$ by $\beta_1 + 2\beta_2x_1$ in an accelerating rate as $x_1$ goes further from its maximum.
The same logic you can apply for $\beta_2 > 0$ while in this case you'll have "smiling" parabola.