**Question**

**Introduction**

This
question is taken from the Haese & Harris textbook for IB HL Mathematics
and is rather challenging. Here I
present two solutions. In the first
solution, I use substitution to make a variable “disappear”. [ heuristics: H10. Simplify the problem, H11. Solve part
of the problem] And that allowed me to crack the rest of them
problem. In the second solution, I
rephrased the problem in terms of tangents to curves. [
heuristic H09. Restate the problem in another way ] This gives another
angle from which to tackle the problem.

**Solution 1**
Both solutions are related in
the sense that they hinge on some form of the relation maked as [*]. That
led to an equation in *a*. Once *a* is found, *k* can be found, and the solutions proceed
similarly. The equation [*] is a manifestation of the fact that for a repeated
root, both *P*(*x*) and *P*’(*x*) share a common factor.

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