Determine the nodes $x_{1}, x_{2}$ of the two-point Gaussian quadrature

$$\int_{0}^{1} f(x) w(x) d x \approx a_{1} f\left(x_{1}\right)+a_{2} f\left(x_{2}\right), \quad w(x)=x$$

and express the coefficients $a_{1}, a_{2}$ in terms of $x_{1}, x_{2}$. [You don't need to find numerical values of the coefficients.]