Why does $2+2=5$ in this example?

Question

I stumbled across the following computation proving $2+2=5$

Clearly it doesn't, but where is the mistake? I expect that it's a simple one, but I'm even simpler and don't really understand the application of the binomial form to this...

Answer 1

The error is in the step where the derivation goes from $$\left(4-\frac{9}{2}\right)^2 = \left(5-\frac{9}{2}\right)^2$$ to $$\left(4-\frac{9}{2}\right) = \left(5-\frac{9}{2}\right)$$

In general, if $a^2=b^2$ it is not necessarily true that $a=b$; all you can conclude is that either $a=b$ or $a=-b$. In this case, the latter is true, because $\left(4-\frac{9}{2}\right) = -\frac{1}{2}$ and $\left(5-\frac{9}{2}\right) = \frac{1}{2}$. Once you have written down the (false) equation $-\frac{1}{2} = \frac{1}{2}$ it is easy to derive any false conclusion you want.

Answer 2

Because $x^2=y^2\Leftrightarrow x=y$ is not true.

Also $\sqrt{x^2}=x$ is wrong: actually, $\sqrt{x^2}=|x|$.

Answer 3

$9^2 = (-9)^2$
But $9 \ne -9$

If we had taken as $\left|4 - \dfrac{9}{2} \right|=\left|5 - \dfrac{9}{2} \right|$
$0.5=0.5$