= Solution
Choose the <Fourier series> convention
$$
V(x)=\sum_{j\in\mathbb Z}V_j e^{ijG_1x},\qquad
V_j=\frac1a\int_0^aV(x)e^{-ijG_1x}\,dx.
$$
For a real <potential energy>, $V_{-1}=V_1^*$. The symbol $V(G_1)$ means the <Fourier coefficient> $V_1$, with units of <energy>, rather than the pointwise value of $V(x)$ at a position $G_1$.
Retain the two <plane waves> $|1\rangle=L^{-1/2}e^{ikx}$ and $|2\rangle=L^{-1/2}e^{i(k-G_1)x}$ in the <nearly-free electron model>. Their kinetic <energies> are $\epsilon_1=\hbar^2k^2/(2m)$ and $\epsilon_2=\hbar^2(k-G_1)^2/(2m)$. The <matrix representation of a linear map> for the <Quantum Hamiltonian> projected onto this two-dimensional <basis> is
$$
H=\begin{pmatrix}\epsilon_1+V_0&V_1\\V_1^*&\epsilon_2+V_0\end{pmatrix}.
$$
Thus $\det(H-EI)=0$ gives
$$
(E-V_0-\epsilon_1)(E-V_0-\epsilon_2)-|V_1|^2=0.
$$
Solving this <quadratic equation> yields
$$
\boxed{E_\pm=V_0+\frac{\epsilon_1+\epsilon_2}{2}
\pm\frac12\sqrt{(\epsilon_1-\epsilon_2)^2+4|V_1|^2}.}
$$
Choosing the zero of <potential energy> so that its spatial average is $V_0=0$ reproduces the requested expression. If that average has not been removed, both <energy bands> are shifted by $V_0$; no <band gap>, <group velocity> or <effective mass> is changed by this constant shift.
The two kinetic <energies> coincide at $k=G_1/2$. Their coupling gives an <avoided crossing> with <band gap> $2|V_1|$. The result is exact within the projected two-wave <Quantum Hamiltonian>. It is an approximation to the full crystal problem, appropriate near this crossing for a weak <periodic potential> whose other <Fourier coefficients> do not cause strong mixing with remote <plane waves>. Small $|V_1|$ alone would not justify ignoring arbitrarily large other <Fourier coefficients>.
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