Suppose the roots of 2x^2 + 10x + 1 are a and B. Suppose another quadratic, x^2 + 1x + r, has roots y = a + 1/B and σ = B + 1/a. What are q and r?
Using the quadratic formula,
a = -1/2 (5+√23)
b = -1/2 (5-√23)
a + 1/b = -3/2 (5+√23)
b + 1/a = -3/2 (5-√23)
(x-(-3/2 (5+√23)))(x-(-3/2 (5-√23))) = x^2+15x+(9/2)
Or, using the roots in terms of a and b,
x^2+x+r = (x-(a+1/b))(x-(b+1/a))
= x^2-((a+1/b)+(b+1/a))x + (a+1/b)(b+1/a)
= x^2 - (a+b+1/a+1/b)x + (ab+1/(ab)+2)
= x^2 + 15x + 9/2
q = 15
r = 9/2
I can't help but feel there is a more elegant solution, but it eludes me at the moment.