 Solution

A real square on the left requires $x^5\ge 0$, so $x\ge 0$. Taking the two square roots gives exactly the two real graph branchesThey meet only at the origin. The local coordinate change $Y=y-x^2$ turns the equation into $Y^2=x^5$, the plane cusp of type (2,5); both branches share the tangent $y=0$ there.