Find the points that have the common tangent line

There are two points on the curve y = x^4 - 2x^2 - x that have a common tangent line. Find those points.

So far I think I know what needs to be found but I cannot put the pieces together.

y-prime = 4x^3 - 4x - 1

i need a y prime that yields a slope such that the equation y=mx+b will intersect the initial curve such that the new x value will give the same derivative value as the first point (thus the same sloped tangent)

but i do not know how to do this

Re: Find the points that have the common tangent line

Sorry for bumping such an old thread – I was solving a similar problem on another forum when I actually found this thread on Google.

Here’s how I do it.

Let . If the equation of the common tangent is we have

We want to find two distnct points at and such that

and

We will show that . If then ; substituting into gives ; also and so . Hence are roots of the quadratic equation , i.e. . We don’t want this as must be distinct.

Therefore . Substituting into gives , i.e. .