TITLE: How many solutions to the rational equation?
QUESTION [6 upvotes]: If $a$ and $b$ and $c$ are parameters, how many solutions for:
$$\frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-a)(x-c)}{(b-a)(b-c)}  + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1$$

I would say $3 \implies x = \{a, b, c\}$
But the answer is apparently $\infty$.
I am confused?

REPLY [2 votes]: You are finding a second-degree polynomial whose graph goes through $(a,1),(b,1),(c,1)$ by Lagrange's interpolation. Such polynomial is unique, and it is clearly the constant $1$.