The image produced by a concave mirror

Using the mirror formula, $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$.

For a concave mirror, the focal length $f$ is negative. Let $f = -F$ where $F > 0$.

If the object is virtual, the object distance $u$ is positive. Let $u = U$ where $U > 0$.

Substituting these into the mirror formula, we get:

$\dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u} = -\dfrac{1}{F} - \dfrac{1}{U} = -$\left(\dfrac{1}{F} + \dfrac{1}{U}\right)

Since both $F$ and $U$ are positive, \dfrac{1}{v} is negative, which implies $v < 0$.

A negative image distance corresponds to a real image formed in front of the mirror. Thus, a concave mirror always forms a real image for a virtual object.