Show that the roots of the equation x^2+px+q=0 are rational,if p=k+q/k,where p,q,k are rational.
For the roots of a quadratic equation ax^2 + bx + c = 0 to be rational b^2 - 4ac >=0
We have the equation x^2 + px + q = 0
If p = k + q/k and the equation has rational roots:
p^2 - 4q
=> ( k + q/k) - 4q
=> k^2 + q^2/k^2 + 2q - 4q
=> k^2 + q^2/k^2 - 2q
=> k^2 - 2q + q^2/k^2
=> (k - q/k)^2
As (k - q/k)^2 is a square it is greater than or equal to 0, which implies that p^2 - 4q >=0.
Therefore the equation x^2+px+q = 0 has rational roots if p=k+q/k.