The best explanation is:
The square root of terms separated by addition and subtraction cannot be calculated individually.
In mathematical expressions, the square root of a sum or difference of terms, such as [tex]\( b^2 - 4ac \)[/tex], cannot simply be separated into the individual square roots of each term and then added or subtracted. For example, [tex]\(\sqrt{b^2 - 4ac}\)[/tex] is not equal to [tex]\( \sqrt{b^2} - \sqrt{4ac} \)[/tex]. Each term inside the square root interacts with each other according to the rules of algebra, and separating them would violate these rules.
In the context of the quadratic formula, we have:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(\sqrt{b^2 - 4ac}\)[/tex] is taken as the square root of the entire expression [tex]\(b^2 - 4ac\)[/tex]. Attempting to rewrite [tex]\(\pm \sqrt{b^2 - 4ac}\)[/tex] as [tex]\( \pm (b \pm \sqrt{-4ac}) \)[/tex] would be incorrect because it implies breaking up the square root across addition/subtraction inside it, which is algebraically invalid. Therefore, the explanation emphasizes the correct treatment of the square root over a sum or difference of terms.
