3-SAT to Max-2-SAT Reduction
Hint: Replace a singleton clause $a$ with $a \lor b$ and many copies of $\lnot b \lor c$, $\lnot b \lor \lnot c$.



Reductions without extra variables don't work even when singleton clauses are allowed. There are five types of clauses: $\ell_1,\lnot \ell_1, \ell_1 \lor \ell_2$, $\ell_1 \lor \lnot \ell_2$, $\lnot \ell_1 \lor \lnot \ell_2$. Suppose that $k$ of $\ell_1,\ell_2,\ell_2$ are true. The probability that a random clause of each type is true is:
$$
\begin{array}{c|cccc}
k&amp;0&amp;1&amp;2&amp;3\\\hline
\ell_1 &amp; 0 &amp; 1/3 &amp; 2/3 &amp; 1 \\
\lnot \ell_1 &amp; 1 &amp; 2/3 &amp; 1/3 &amp; 0 \\
\ell_1 \lor \ell_2 &amp; 0 &amp; 2/3 &amp; 1 &amp; 1\\
\ell_1 \lor \lnot \ell_2 &amp; 1 &amp; 1/3 &amp; 1/3 &amp; 1\\
\lnot \ell_1 \lor \lnot \ell_2 &amp; 1 &amp; 1 &amp; 2/3 &amp; 0
\end{array}
$$
We would like to take a convex combination of these rows which is of the form
$$ \begin{array}{cccc} \alpha &amp; \beta &amp; \beta &amp; \beta \end{array} $$
However, all such convex combinations have $\alpha = \beta$.