None of the answer is correct for all possible quadratic equations. Here is a counter example.
For $x^{2} - 2x + 1 = 0$, the only root is $1$ and it satisfies the criteria of being real & positive. Also the other criteria of $a, b, c$ being real numbers is satisfied as well.
Now, the equation $x^2 -2|x| + 1 = 0$ can be defined as two equations $x^{2} - 2x + 1 = 0$, $x \geq 0$ and $x^{2} + 2x + 1 = 0$, $x < 0$ which only have a total of 2 roots which are $1$ and $-1$ respectively and not 4 roots.
All quadratic equations where the discriminant is zero and $b$ & $a$ are of same sign will only give 2 roots for such transformation.