At what angle do these curves cut one another? I'm working on an exercise that asks this: At what angle do the curves $$y = 3.5x^2 + 2$$ and $$y = x^2 - 5x + 9.5$$ cut one another? I have set these equations equal to one another to find two values for x. Namely, $x = 1$ and $x = -3$ as intersections. How should I proceed? Most everyone here is always extremely helpful so I can't thank you all enough in advance for any assistance.

You know the curves cut themselves at $x=1$ and $x=-3$. Let's consider a general case you might find helpful. Consider two functions $f,g$ that intersect at a point $x=\xi$. 
Consider now the tangent line of $f$ at $x=\xi$. What angles does it make with the $x$ axis? It shouldn't be new news that $\tan\theta=f'(\xi)$. Similarily, the tangent line of $g$ at $x=\xi$ makes an angle $\eta$ with $\tan\eta =g'(\xi)$. So at what angle do these lines cross? Would you be convinced it must be $\rho=\theta-\eta$? You can make a drawing, and of course choose $\theta$ to be the largest angle. But we know that $$\tan(\theta-\eta)=\frac{\tan\theta-\tan\eta}{1+\tan\theta\tan\eta}$$
That is, $$\tan(\rho)=\frac{f'(\xi)-g'(\xi)}{1+f'(\xi)g'(\xi)}$$
You should be careful about the angles you're dealing with so as not to get any "off-set" results!
