The given vertex is (0,0) and
the parabola is symmetric about the y - axis.
Hence the equation of the parabola is either of the form.
$x^2=4ay $ or $ x^2 = -4ay$
The curve passes through the point (5,2) .
Now substituting for x and y we get,
$(5)^2=-4a(2)$
$ \Rightarrow 25 = -8a$ or
$a = -\large\frac{25}{8}$
Hence the equation of the parabola is
$ x^2=-4 \bigg( \large\frac{25}{8} \bigg)$$y$
(i.e) $x^2=-\large\frac{25}{2}$$y$
or $2x^2=-25y$ is the required equation of the parabola.