Question

Suppose a and b are the roots of the quadratic equation x^2 +3x +5=0.
What is the monic quadratic equation with roots a + 1/a and b +1/b?

Answer 1

Just use your sum and product of roots of a quadratic property.
Here a+b = -3 and ab = 5

sum of new roots = a + 1/a + b +1/b
= a+b + 1/a + 1/b
= a+b + (b+a)/ab = -3 + -3/5 = -18/5

product of new roots = (a + 1/a)(b +1/b)
= ab + a/b + b/a + 1/ab
= 5 + (a^2 + b^2)/ab + 1/ab
= 5 + (a^2 + b^2)/5 + 1/5

So what is a^2 + b^2 ???
well, (a+b)^2 = a^2 + 2ab + b^2
so a^2 + b^2 = (a+b)^2 - 2ab = 9 - 10 = -1

so back to
5 + (a^2 + b^2)/5 + 1/5
= 5 + -1/5 + 1/5 = 5

your new equation is
x^2 + 18/5 x + 5 = 0
or 5x^2 + 18x + 25 = 0