The parabola $y= 4px^2$ has vertex at (0, 0) and focus at (0, p). The circle with center (0, p) and radius p has equation $x^2+ (y- p)^2= p^2$. Those two graphs intersect when $y= 4px^2= p- \sqrt{p^2- x^2}$. Write that as $\sqrt{p^2- x^2}= p- 4px^2$ and square both sides: $p^2- x^2= p^2- 8p^2x^2+ 16p^2x^4$ or $16p^2x^4- (8p^2- 1)x^2= x^2(16p^2x^2- (8p^2- 1))= 0$. That has x= 0 as a double root and $x= \pm\sqrt{\frac{8p^2- 1}{16p^2}}$ are the other two roots.