Are there more quadratics with real roots or more with complex roots? Or the same?

Consider all quadratic equations with real coefficients:

$$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0 $$

I was wondering whether if more of them have real roots, more have complex roots, or a same number of each?

I thought about this graphically, and wlog considered only the ones with positive $a$. By categorising the quadratics by their minimum point, there are an equal number of possible minima below the $x$-axis as above, so it seems that there are an equal number of quadratics with real roots as complex ones.

However, I then considered this problem algebraically by using the discriminant:

$$b^2-4ac$$

If $a$ and $c$ have opposite signs, then the discriminant will be positive, and so the quadratic will have real roots. This represents $50\%$ of the quadratic equations possible.

However, if $a$ and $c$ have the same sign, then some of these quadratics have real roots, whilst others have complex roots depending on whether $4ac$ is bigger than $b^2$ or not.

This suggests that there are more quadratics with real roots which contradicts the above answer.

Is the reason I've reached contradicting answers something to do with infinites, and that I can't really compare them in the way I've done above?