In the book of Lee, introduction to smooth manifolds, in page 123, it is asked that
\begin{array}{c}{\text { For each } a \in \mathbb{R}, \text { let } M_{a} \text { be the subset of } \mathbb{R}^{2} \text { defined by }} \\ {M_{a}=\left\{(x, y) : y^{2}=x(x-1)(x-a)\right\}} \\ {\text { For which values of } a \text { is } M_{a} \text { an embedded submanifold of } \mathbb{R}^{2} ? \text { For which }} \\ {\text { values can } M_{a} \text { be given a topology and smooth structure making it into an }} \\ {\text { immersed sub manifold ? }}\end{array}
I proceeded as defining $F(x,y) = x(x-1)(x-a) - y^2$, and observing that $$DF(x,y) = (3x^2 +2(1+a)x -a \quad -2y)^T.$$ If we can show that $F(x,y) = 0$ is a regular value for $F$, then we can argue that $M_a$ is an embedded smooth manifold.
$DF$ has to have rank $1$, so if $y\not =0$, then nothing to check. If $y= 0$, then we need $3x^2 +2(1+a)x - a\not = 0$ for all $x$ satisfying $x(x-1)(x-a) = 0$, but the latter equation is satisfied only for $x=0, 1, a$.Therefore, $x=0 $ implies $a\not =0$, $x=1$ implies $a\not = 1$, and $x=a$ implies $a \not = 1, 0$. Hence, for any value of $a$ except $0,1$, $M_a$ is an embedded sub manifold of $R^2$ with codimension $1$.
Question:
Is my solution correct ? Am I missing something ?