$2+2 = 5$? error in proof $$\begin{align} 2+2 &= 4 - \frac92 +\frac92\\
&= \sqrt{\left(4-\frac92\right)^2} +\frac92\\
&= \sqrt{16 -2\times4\times\frac92 +\left(\frac92\right)^2} + \frac92\\
&= \sqrt{16 -36 + \left(\frac92\right)^2} +\frac92\\
&= \sqrt {-20 +\left(\frac92\right)^2} + \frac92\\
&= \sqrt{25-45 +\left(\frac92\right)^2} +\frac92\\
&= \sqrt {5^2 -2\times5\times\frac92 + \left(\frac92\right) ^2} + \frac92\\
&= \sqrt {\left(5-\frac92\right)^2} +\frac92\\
&= 5 + \frac92 - \frac92 \\
&= 5\end{align}$$
Where did I go wrong
 A: Basically, your proof is saying
$$
-0.5=\sqrt{(-0.5)^2}=\sqrt{0.5^2}=0.5
$$
Now find the error.
A: In the first line you have $4-4.5=\sqrt{(4-4.5)^2}$, which isn't true, because $-0.5\neq 0.5$.
A: Apart from the other answers, even at the last,  $\sqrt{(5-\frac92)^2}=\pm(5-\frac92)$. with + it is wrong.
With $-(5-\frac92)$, that is $-5+\frac92$, adding the other $\frac92$ from the original equation, we do get $4$.
A: $\sqrt{\left(4 - \frac 9 2 \right)^2} = 4 - \frac 9 2 = -0.5.$
It's not true.
If $a \geq 0$, then $\sqrt{a} \geq 0$.
A: How did $\sqrt{16 -2\times4\times\frac92 +(\frac92)^2}$ turn into $\left(\sqrt{16 -36 + (\frac92)^2}\right)^2$?
Then later, you seem to assume that since $\left(4-\frac92\right)^2$ is the same as $\left(5-\frac92\right)^2$, it follows that $4-\frac92=5-\frac92$.  Like saying that since $3^2=(-3)^2$, it follows that $3=-3$.  A well known mistake.
A: Here's what your "proof" would look like correcting all the errors. As you can see, it's not nearly as impressive as a proof that 2+2=5.
$$\begin{align} 2+2 &= 4 - \frac92 +\frac92\\
&= -\sqrt{(4-\frac92)^2} +\frac92\\
&= -\sqrt{16 -2\times4\times\frac92 +(\frac92)^2} + \frac92\\
&= \left(-\sqrt{16 -36 + (\frac92)^2}\right) +\frac92\\
&= \left(-\sqrt {-20 +(\frac92)^2}\right) + \frac92\\
&= -\sqrt{25-45 +(\frac92)^2} +\frac92\\
&= -\sqrt {5^2 -2\times5\times\frac92 + (\frac92) ^2} + \frac92\\
&= -\sqrt {(5-\frac92)^2} +\frac92\\
&= -5 + \frac92 + \frac92 \\
&= -5+9\end{align}$$
For reference, the most serious mistake was in the 2nd line. In general, it's not true that $\sqrt{x^2} = x$, but rather $\sqrt{x^2} = |x|$. For $x=4-\frac92<0$, you need to keep track of the extra minus sign coming from the absolute value. Other than that, there were some obvious typos that I've corrected.
A: $$\sqrt{a^2}=|a|\not=a$$
Watch the signs.
A: In the very first line, you assumed that $a=|a|=\sqrt{a^2}.$ But this is not true in general. In particular, it is false for negative $a.$ And in fact, $$4-\frac92<0,$$ so in this case we do not have that $$4-\frac92=\sqrt{\left(4-\frac92\right)^2}=\left|4-\frac92\right|$$ as assumed.
A: Clearly, $\sqrt{(4-\frac{9}{2})^2}$ is what? Well, $4-\frac{9}{2}$ is $-\frac{1}{2}$. But we all know $(-\frac{1}{2})^2$ is $\frac{1}{4}$ since $(-x)^2=x^2$. That means that $\sqrt{(4-\frac{9}{2})^2}$ is $\frac{1}{2}$. So the assumption that $\sqrt{(4-\frac{9}{2})^2} = 4 - \frac{9}{2}$ leads to $-\frac{1}{2}=\frac{1}{2}$ which is clearly wrong.
So the square root of a positive number squared is a positive number and the square root of a negative positive number squared is also a positive number. So if we assume $\sqrt{(-x)^2} = -x$ to $x = -x$ which is false.  Note: x is positive, not negative. This is the correct statement:
$\sqrt{(-x)^2} = \sqrt{x^2} = x \not= -x$ 
