[Math] Prove rational sum and product of two irrational numbers

Question

I need to prove that

$$\exists a,b \in \mathbb{R} \setminus \mathbb{Q} : a + b, ab \in \mathbb{Q}$$

Any ideas? I, unfortunately, don't have one yet. The most obvious way with equations in integers (using the definition of a rational number) with irrational coefficients doesn't seem to bear any fruit.

Answer 1

Hint. Think about the quadratic formula for a quadratic equation with integer coefficients but two irrational roots.