<s>[INST] Find the solutions of the equation $\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-a)(x-c)}{(b-a)(b-c)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=3x-2$ 
Find the solutions of the equation $$\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-a)(x-c)}{(b-a)(b-c)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=3x-2$$

If we consider the expression as an identity and substitute $x=a$ we get the value of $a=1,2$ similarly $b=1,2$ and $c=1,2$
If we substitute these values into the expression at least one of the denominator will be equal to $0$ which is not possible therefore this expression is not an identity. How to proceed further?
 [/INST] Assume $a,b,c$ are fixed distinct (all different) real number values. (Otherwise the left-hand side is undefined.)
Then the left-hand side is a quadratic function in $x$. Call this function $f(x)$.
We have $f(a) = a^2$, $f(b)=b^2$ and $f(c)=c^2$. Since a quadratic is determined by its values at three points, we must have $f(x)=x^2$. (This expression for $f(x)$ should be independent of the values of $a, b$, and $c$, provided, as noted above, that these values are all different.)
So the equation becomes $x^2=3x-2$, which you can now easily solve for $x$.
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