The equation of a parabola can be written y=a(x-h)²+k where (h,k) is the vertex. The focus is at (h,k+p) and the directrix line is y=k-p where ap=1/4.
So (0,-5)=(h,k+p) making h=0, -5=k+p and 5=k-p.
From this we get k=0 and p=-5, making a=-1/20.
The equation is y=-x²/20.