Help needed in determining the singularity

Question

Can someone teach me how to determine the singularity of algebraic curve $y^2 =x^3+x^2$. I'll be really grateful and thanks in advance

Answer 1

Let $f(x,y)=y^2-x^3-x^2$. Make $f(x,y)=0$, $\dfrac{df}{dx}=0$ and $\dfrac{df}{dy}=0$. The singularities are the common points of the equations.