The following operations are permitted with the quadratic polynomial
px

^{2}+qx+r

- Switch p and r
- Replace x by x + s, where s is any real number.

By repeating these operations, can the quadratic expression x

^{2} - x - 2 be transformed into the expression x

^{2} - x – 1?

The first operation replaces the roots with their reciprocals. The second is a simple shift on the number line. Note that only the first operation changes the difference between the roots.

I assert this without proof: For any pair of positive real numbers m and n, there is a pair of quadratics f(x) and g(x) such that the roots of f(x) are the reciprocals of g(x), the difference between the roots of f(x) is m, and the difference between the roots of g(x) is n.

Then I construct f(x) and g(x) based off of the two given polynomials. So I use three operations to change x^2-x-2: first Operation 2 changes x^2-x-2 to f(x), second Operation 1 changes f(x) to g(x), and third Operation 2 changes g(x) to x^2-x-1.

But this may easily make the final result a multiple of x^2-x-1. It might be possible to multiply f(x) by a suitable coefficient to get the desired result.