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$$ax^2 + bx + c = 0$$ is a quadratic equation, $$a$$, $$b$$, $$c$$ are rational numbers. $$\Delta = b^2 - 4ac$$.
The roots are $$\frac{-b \pm \sqrt{\Delta}}{2a}$$. If $$\Delta = k^2$$ with $$k$$ rational, $$\sqrt{\Delta} = |k|$$ and both roots are rational. If $$\Delta$$ is not a rational square, the roots are irrational conjugates (or non-real when $$\Delta < 0$$), so a, c and d fail. Only option b is correct.
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