Question

Find 2 numbers a and b such that (a-b)^2 < (a+b)(a-b)< (a+b)^2 Find two numbes a and b such that this is not true

Answer 1

Suppose for the moment that the inequality holds for all $a,b$:

$(a-b)^2$

Expanding everything gives

$a^2-2ab+b^2$

$\implies-ab$

In particular, the inequality says that $-ab$ for any choice of $a,b$. But if $a$ and $b>0$, then $-ab>0$ while $ab$.

So one possible choice of $a,b$ could be $a=-1$ and $b=1$. Then we get

$(-1-1)^2=4$

$(-1+1)(-1-1)=2$

$(-1+1)^2=0$

but clearly it's not true that $4$.