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An $n$-node Gaussian quadrature formula chooses nodes and weights for an integral with a positive weight so that every polynomial of degree at most $2n-1$ is integrated exactly. Its nodes are the zeros of the degree-$n$ orthogonal polynomial for that weight.

For the even weight $w(x)=1-x^2$, the monic orthogonal quadratic has the form $p_2(x)=x^2-d$. Orthogonality to $x$ follows from oddness; orthogonality to one givesThe nodes are $\pm 1/\sqrt{5}$. The equal weights sum to $4/3$, hence each is $2/3$. The two-node Gaussian quadrature with quadratic weight is thereforeTo verify cubic exactness, divide any cubic as $P=p_{2}Q+R$ with $\mathrm{deg}Q,\mathrm{deg}R\le 1$. Orthogonality removes $p_{2}Q$ from the integral and its nodal values vanish; linear interpolation integrates $R$ exactly. This also proves the Gaussian property rather than merely matching two moments.