If b is rational and c is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only definition of rational numbers to prove.

Question

If $b$ is rational and $c$ is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only the definition of rational numbers to prove.

I think that it can't because since $b$ is rational, if two real roots exist then both are either rational or irrational, however, I'm not sure how to prove this using only the definition of rational numbers i.e. it is a number that can be expressed as $\frac{x}{y} $ where $x$ and $y$ are integers $y$ being nonzero. So you can't assume stuff like rational + irrational = irrational.

Answer 1

Of course, note that all rational numbers are indeed real.

---

I suppose you wanted to ask "Is it possible that one root is rational and the other is not". The answer to that is then "no".

Note that the sum of the two roots is $-b \in \Bbb Q$. Using the definition of rationals, it follows that the difference of two rationals is rational. Conclude that $\text{rational} + \text{irrational} = \text{rational}$ is not possible.

Answer 2

Hint

Suppose $r$ is a rational root and $j$ is a irrational root.

$$r+j=-b\to j=-(b+r).$$

Can you see the problem?