Solution
= Solution
A two-node Gaussian rule is exact through degree three, the maximal degree $2n-1$. Its nodes are the zeros of the monic quadratic orthogonal to $1,x$ for weight $x$ on $[0,1]$. Writing it as $p=x^2+ux+v$ gives
$$
\int_0^1px\,dx=0,\qquad\int_0^1px^2\,dx=0,
$$
so $u=-6/5$, $v=3/10$. Therefore
$$
\boxed{x_0=\frac35-\frac{\sqrt6}{10},\qquad
x_1=\frac35+\frac{\sqrt6}{10}.}
$$