 \iff x(x - 5) = 0 \\
    &amp; \iff x \in \{ 0, 5 \}.
  \end{align*}
</p>
<h2 id="mathematical-fallacy">Mathematical Fallacy</h2>
<p>
  Often fallacies or spurious proofs of contradictions is formed by
  sneaking such a division by \( 0 \) into the proof.  Here is one I
  such spurious proof:
</p>
<p>
  Let \( a = b \).  Then

  \begin{align*}
    a + b = 2a
    &amp; \iff a + b - 2b = 2a - 2b \\
    &amp; \iff a - b = 2(a - b).
  \end{align*}

  Dividing both sides by \( a - b \), we get

  \[
    1 = 2.
