How do I test coefficient of a variable equals joint coefficient of a quadratic variable Let us say I have a regression equation such as:
$E(y) = \beta_{0} + \beta_{1} \cdot \text{var}_1 + \beta_{2} \cdot \text{var}_2 + \beta_{3} \cdot \text{var}_2^2$
How do I test if the effect of var1 equals the effect of var2? I know how to do it if there is no quadratic term
 A: I'll work on the assumption that you want to test whether the derivatives with respect to time are the same:
$\text{H}_0: \beta_1 = \beta_2 + 2\beta_3 x\quad$ for all $x$
This will be true for every $x$ if simultaneously $\beta_3=0$ and $\beta_2-\beta_1=0$. If $\beta_3\neq 0$ then it could only be true at one value of $x$ and if $\beta_3=0$ but $\beta_2-\beta_1\neq 0$ then it won't be true at any value of $x$.
Which is to say it's the same as:
$\text{H}_0: \beta_1 = \beta_2\: \text{ & }\: \beta_3 =0$ for all $x$
Since these are linear restrictions the standard partial F approaches could be applied. 
You can either (i) fit the full model with all variables free and the reduced model where both restrictions apply and take 
$F = \frac{\text{"change in RSS"}/2}{RSS_{\text{full}}/dfe_\text{full}}$
(here "RSS" would be "residual sum of squares" and "dfe" is "degrees of freedom for error")
or (ii) you can also do it by reparameterizing the full model to be in terms of a common linear coefficient and the difference and then an F-test can be constructed from the table of sums of squares of the reparameterized full model.
[I don't know that it really makes sense to do this in the particular context, but I'll take the question on its face and leave any potential issues with that for others to comment on.]
