Question analysis so amazing

Question

How this process has been calculated in a manner added.

Answer 1

Note that $$4-\frac 92\not=\sqrt{\left(4-\frac 92\right)^2}$$

Answer 2

$$\sqrt{\left(4-\frac92\right)^2}=-\left(4-\frac92\right)$$

More generally for real $a,$ $$\sqrt{a^2}=|a|=\begin{cases} a &\mbox{if } a\ge0 \\ -a & \mbox{if } a<0\end{cases}$$