Why this value is not the second root of the equation? $\sqrt{x+3}=x-2$
Why $\frac{5}{2} - \frac{\sqrt{21}}{2}$ is not root?
There is only one restriction: $\sqrt{x+3}$, but $\frac{5}{2} - \frac{\sqrt{21}}{2} > 0$.
$x^2-4x+4=x+3$,   $x^2-5x+1=0$,   $D=25-4=21>0$
$D>0$, =>, $x = \frac{ 5 \pm \sqrt{21} }{ 2 }$

That is because the irrational equation:
$$\sqrt A=B\iff A=B^2\;\textbf{ and }\, B\ge 0,$$
i.e., in the present case, $\;x\ge 2$, whereas $\;\dfrac{5-\sqrt{21}}2<\dfrac12$.
