Question

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

Answer 1

Yes, a quadratic equation with rational coefficients may have irrational roots.

Example: Consider the following equation,

x² + 3x + 1 = 0

The roots of the given equation are \(\frac{(-3 \pm \sqrt5)}{2}\)  which is irrational.