So I am given a curve, $E$, with equation: $$ f(x,y) = y^2 - x^3 - ax - b = 0 $$ and I have to prove that if the discriminant $\Delta = 4a^3 + 27b^2 = 0$ then E is singular.
Because I have been given this information as an assumption throughout the year, I am struggling to understand a way to prove this.
HINT. You want to show the point at infinity is never singular (just homogenize and then take the $z$-partial). Otherwise, your curve will have a singular point if and only if $x^3+Ax+B$ has a repeated root if and only if the discriminant is $0$. Compute the discriminant of your polynomial in terms of $A,B$, what do you get?