For a given function $f$ let define a quadrature $$Q(f) = \alpha f(a) + \beta f(c),$$
and an integral: $$I(f) = \int_a^b f(x) \, dx$$
Equations $Q(f) = I(f)$ for monomials $f = 1,\,x,\,x^2$ are
$$\beta+\alpha = b-a, \quad \beta c+a \alpha = b^2/2-a^2/2, \quad \beta c^2+a^2 \alpha = b^3/3-a^3/3$$
The system has a solution: $$ \alpha = (b-a)/4, \quad \beta = 3(b-a)/4, \quad c = (2b+a)/3 \tag{*}\label{*} $$
Unfortunately $Q(x^3) = I(x^3)$: $$ \beta c^3+a^3 \alpha = b^4/4-a^4/4 $$
and \eqref{*} are inconsistent. For example for $a=0$ and $b=1$ it reduces to $2/9 = 1/4$. So the quadrature is accurate for polynomials up to degree $3$. For $a=0$ and $b=1$ it is: $$ Q_0(f) = \frac{f(0) + 3 f(2/3)}{4} $$