Orthogonal polynomials, Legendre

We can proceed in a similar fashion in order to determine the coefficients of \( L_2 \) $$ \begin{equation*} L_2(x) = a+bx+cx^2, \end{equation*} $$ using the orthogonality relations $$ \begin{equation*} \int_{-1}^1L_0(x)L_2(x)dx=0, \end{equation*} $$ and $$ \begin{equation*} \int_{-1}^1L_1(x)L_2(x)dx=0, \end{equation*} $$ and the condition \( L_2(1)=1 \) we would get $$ \begin{equation} L_2(x) = \frac{1}{2}\left(3x^2-1\right). \tag{12} \end{equation} $$