algebra precalculus - Why does $2+2=5$ in this example?

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

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.