The focal length of a concave mirror is negative and that of a convex mirror is positive. You can also prove the same by using the mirror formula (1/f = 1/v +1/u).
Let's see how,
Since we know object is always placed on the left side or direction opposite to the incidence ray of the mirror. Therefore object distance will always be negative.
u = -u
v = -v (Image distance is negative, since image formed by concave mirrors are generally on the left side or direction opposite to the incidence ray)
Using mirror formula,
1/f = 1/v +1/u
or, 1/f = (-u-v)/uv
or, f = -uv/(u+v)
As you can see, The value of f is negative here…
However v is not always negative in concave mirror, when object is placed between focus and pole, you will get a virtual image on the right side of the mirror, so in this case image distance will be positive. But even in this case focal length will be negative. How?
Just use the mirror formula again, and you will know how.