How do I determine the x-coordinates of the points of intersection of a tangent line and the parabola y = x^2?

Question

So in the following problem I have completed part a, but I am stuck on part b. I am trying to find the points of intersection of the tangent line I found in part a and the parabola y = x^2. However, I somehow got stuck in trying to solve part b and I am not sure if I made an obvious mistake. Can anyone point me in the right direction? Thank you!

Answer 1

Part a) seems fine. For b), you are on the right track, simply solve $x^2 - 2ax + (a^2-1) = 0$ or equivalently $x = a+1$ or $x=a-1$.