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 $y^2-x^5$
 A: The points where the curves will intersect is where and only where
$$y^2+yx^2-x^3=y^2-x^5$$
$$\iff yx^2 - x^3 = -x^5 $$ $$\iff x^2(y - x + x^3) = 0$$
$$\iff x = 0 \quad\text{or}\quad y =x(1-x)(1+x)$$
Clearly, $(0, 0)$ is one such point of intersection. 
There are are other points where the curves intersect: indeed, the set of points where the two given "curves" intersect are all points lying on two different curves, both passing through (intersecting at) $(0, 0)$:


*

*The vertical line $x = 0: \{(0, y): y \in \mathbb R\}$. That is, your curves intersect along the y-axis, and clearly $(0, 0) \in x = 0$.

*The curve $y = x(1-x)(1+x) = x - x^3: \{(x, x - x^3): x \in \mathbb R\}$, which also passes through $(0, 0)$.
The graph below doesn't show very well that the two given curves also intersect along the x-axis, but I'll include the graph of all points at which your curves intersect:
$\qquad y^2+yx^2-x^3=y^2-x^5$
$\quad$
A: We will assume that you mean $0 = y^2 + yx^2 - x^3$ and $0 = y^2 - x^5$ and you want to see how many curves it appears is passing through the point $(0,0)$.
Let us perturb $x$ slightly about $0$ and look at how many solutions for $y$ there will be.
For the first curve, we'll have a quadratic with discriminant $$ x^4 + 4 x^3 $$ which will be positive for small positive $x$ but negative for small negative $x$ meaning there will be two solutions for $y$ when $x$ positive and none when $x$ negative.
The second curve will have two solutions for $y$ when $x$ positive but $0$ otherwise.
This means that there will be a total of $2$ curves as you approach $(0,0)$, both coming from the right hand side.
A: 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.
A: Do you mean the intersection number, as in $x^2, x$ intersecting with multiplicity 2? Then (1) has instructions for calculating it. I got 7, but I could have made a mistake.
