Finding points of intersection which are coincident I was practicing some basic math and I faced a question that I have no idea what does it requires me to do. Consider this simple function:
$$f(x) = x^2-5x-12$$
Part a and b where pretty easy. I tried to solve part C as you can see below. Unfortunately I cant imagine what has been asked in part D. What does coincident points of intersection mean? (Literally intersection means cutting each other and Coincident means on top of another AFAIK) Any help/explanation would be appreciated!
Part C:
The point at which the line through $(1,-15)$ with slope $m$ cuts the graph of $f(n)$.
$$m=f'(n)=f'(1)=-1$$
$$y-y_1 = m(n-n_1)=>y+15=-1(n-1)$$
$$ \implies y=-n+1-15$$
Part D:
The values of m such that the points of intersections found in $c$ are coincident.
 A: The slope of the tangent line to $y=2x^2-5x-12$ at $x=1$ is indeed $-1$, and when $x=1$ we have $y=-15$. However, in solving problems like C), it is best not to notice these facts, because you might be asked a similar question in which the given point is not on the curve.  Also, you were asked explicitly to find the intersection points with the curve of the general line with slope $m$ that passes through $(1,15)$, not just the line with specific slpe $m=-1$. 
The line through $(1,-15)$ with slope $m$ has equation $y-(-15)=m(x-1)$, that is, $y=mx-m-15$.  We want to find the point(s) at which this line meets the curve $y=2x^2-5x-12$.
Substitute $mx-m-15$ for $y$ in the equation of the curve.  We get a quadratic equation in $x$, which turns out to be
$$2x^2-x(5+m)+(3+m)=0.$$
Solve. In principle, we use the Quadratic Formula, or complete the square. However, the problem poser has kindly chosen numbers that make the quadratic factor nicely. One root is (as we already knew) $x=1$, and the other is $x=\frac{m+3}{2}$. The two roots coincide when $m=-1$. 
Remark: The problem is presumably intended as an approach to the tangent line that does not use the derivative. And indeed Fermat, among others, found tangent lines to quite a few curves by using precisely the "double root" idea, well before the work of Newton and Leibniz on the calculus. 
