(These are the solutions from a student of mine, whose identity is private. I will call him, RI, Bengaluru.)

**Question I: **

Solve as elegantly as possible:

**Solution I (of RI, Bengaluru):**

Hence, .

**Question II: **

Find the necessary and sufficient conditions on the coefficients p, q, and r of the given cubic equation such that the roots of the cubic are in AP:

**Solution II: (credit to RI, Bengaluru) : **

Let the roots of the above cubic be .

By Viete’s relations: and and so that we get so that and from the second equation we get , that is, so that , and exploiting the third Viete’s relation we get , that is , which is the required necessary and sufficient condition, where .

**Method II: ***For pedagogical purposes. The above solution to question 2 was quick and elegant because of the right choice of three quantities in AP *as . Do you want to know how ugly and messy it can get if a standard assumption is made:

Let the roots of the cubic be . Let d be the common difference. So that the three roots in AP are , and

Then, applying Viete’s relations, we get so that and which changes to and the third viete’s relation gives us .

The second Viete’s relation is a quadratic in and the third Viete’s relation is a cubic in . This is how messy it can get…at least, you will agree RI’s judicious choice has rendered a clean, quick solution.

Regards,

Nalin Pithwa.

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