Question

Let `alpha` and `beta` be the roots of the equation `x^2+2px-2q^2=0` where `p` and `q` are rational and `p^2+q^2` is not a perfect square, form the quadratic equation whose one root is `alpha+beta+sqrt(alpha^2+beta^2)`?

Answer 1

`x^2+4px-4q^2=0`

Explanation:

As `alpha` and `beta` are roots of `x^2+2px-2q^2=0`, we have

`alpha+beta=-2p` and `alphabeta=-2q^2`

Hence `alpha+beta+sqrt(alpha^2+beta^2)`

= `alpha+beta+sqrt((alpha+beta)^2-2alphabeta)`

= `-2p+sqrt((4p^2+4q^2)`

= `-2p+2sqrt(p^2+q^2)`

As `p` and `q` are rational and `sqrt(p^2+q^2)` is not rational, we must have the other root of the desired equation as its conjugate i.e. `-2p-2sqrt(p^2+q^2)`

then sum of roots is `-4p` and product of roots is `4p^2-4(p^2+q^2)=-4q^2` and

equation is `x^2+4px-4q^2=0`