<s>[INST] 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 $\frac{5}{2} - \frac{\sqrt{21}}{2} > 0$.
$x^2-4x+4=x+3$,   $x^2-5x+1=0$,   $D=25-4=21>0$
$D>0$, =>, $x = \frac{ 5 \pm \sqrt{21} }{ 2 }$
 [/INST] There is actually a second restriction to your equation, which is "hidden":
Since $\sqrt{x+3} \geq 0$ you must also have $x-2 \geq 0$, which your root fails.
Note that the second root actually satisfies $\sqrt{x+3}=-(x-2)$.
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