How many intersections has the following two curves at the point $(0,0)$? The curves are as follows:
$y^2+yx^2-x^3$ and $y^2-x^5$

Let $x=z^2,y=z^5$
$\implies (z^5)^2+z^5\cdot z^2-(z^2)^3=0\implies z^6(z^4+z-1)=0$
So, $z$ has $6$ repeated values for $z=0$, 
So, $x$ will have $\frac62=3$ repeated values as one value $t^2$ of $x$ corresponds to two values of $z$ namely, $\pm t$
If $x=0,y^2=0\iff y=0$ 
So,we will have $3$ intersections.
