Two-node Gaussian quadrature with quadratic weight (${\int }_{-1}^1(1-x^2)f(x)dx\simeq \frac{2}{3}[f(-1/\sqrt{5})+f(1/\sqrt{5})]$)

The even weight $1-x^2$ has moments $m_0=4/3$ and $m_2=4/15$. Its monic quadratic orthogonal polynomial is $x^2-1/5$, so the two Gaussian quadrature nodes are $\pm 1/\sqrt{5}$. Symmetry and constant exactness give weights $2/3$ each. These nodes and weights integrate every polynomial of degree at most three exactly. Degree four fails: the true integral is $4/35$, whereas the rule gives $4/75$.