Question which has no answers Sorry, I don't really know how to improve this question, this quiz has been given me today and I think there is no a true answer, because in the different cases of $a &gt; 0, a &lt; 0, b &gt; 0, b &lt; 0$(and nested cases) you have that every answer can be false.

Taken $a, b \in \mathbb{R}$, given $a^2 - b^2 = 0$, which one is true?


$ab &gt; 0$
$a + b = 1$
$ab &lt; -1$
$a &gt; b$
$a + b = 0$


These are answers I gave myself(all the examples respect the given condition of $a^2 - b^2 = 0$:


If $a = 1 \land b = -1$ then $ab = -1$
If $a = 1 \land b = -1$ then $a + b = 0$
If $a = 0 \land b = 0$ then $ab = 0$
If $a = -1 \land b = 1$ then $a &lt; b$
If $a = 1 \land b = 1$ then $a + b = 2$


Any suggestions? Thanks!
All of the options are false, your couter-examples are okay. Some statements you can get from $a^2 - b^2 = 0, a,b \in \mathbb R$ are:
$$\begin{align*}a^2 &amp; = b^2\\|a|^2 &amp; = |b|^2\\|a| &amp;= |b|\\a&amp;=\pm b\\b&amp;=\pm a\end{align*}$$
since a2 - b2 = 0
by factorization
(a+b)(a-b) = 0

a + b = 0 and a - b = 0

Thus , among all options last one a + b = 0 is correct