Question

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational ? Justify your answer.

Answer 1

Yes, for this Discriminant should not be a perfect square.

Consider the quadratic equation $2x^2+x-4=0$, which has rational coefficients.

The roots of the given quadratic equation are $\frac{-1+\sqrt{33}}{4}and\frac{-1-\sqrt{33}}{4}$, which are irrational.