What is wrong with this proof of: $2+2 = 5$

Question

I have seen this image and surprised that we can prove $2 + 2 = 5$. can any one tell me what is wrong with this image.

Prove that, $2+2=5$.

We know that, $2+2=4$

$$\begin{align}\Rightarrow2+2&=4-\dfrac92+\dfrac92\\\,\\ &=\sqrt{\left(4-\dfrac92\right)^2}+\dfrac92\\ &=\sqrt{16-2\cdot4\cdot\dfrac92+\left(\dfrac92\right)^2}+\dfrac92\\ &=\sqrt{-20+\left(\dfrac92\right)^2}+\dfrac92\\ &=\sqrt{(5)^2-2\cdot5\cdot\dfrac92+\left(\dfrac92\right)^2}+\dfrac92\qquad\qquad\qquad\\ &=\sqrt{\left(5-\dfrac92\right)^2}+\dfrac92\\ &=5-\dfrac92+\dfrac92\\ &=5\\\,\\&\therefore\,2+2=5\text{ (Proved)}\\ \end{align}$$

Answer 1

Hint: Given any real number $x,$ we have $$\sqrt{x^2}=|x|.$$

Answer 2

No,its not possible

your second step is wrong

$4-4.5$ is negative number

So $\sqrt{(4-4.5)^2}$ is not possible

Answer 3

Using this, not only you can prove $4=5$ but you can also prove $a=b$ for all different $a,b$. BUT all these fallacies come from a slip in part of the proof. Here the mistake comes from accepting that $\sqrt x$ can be negative. This is just wrong.

Every time you see such a thing, look at steps of proof and definition of operators carefully.