The best explanation for why the expression [tex]\( \pm \sqrt{b^2 - 4ac} \)[/tex] cannot be rewritten as [tex]\( D \pm \sqrt{-4ac} \)[/tex] during the step after Step 7 is:
The square root of terms separated by addition and subtraction cannot be calculated individually.
Here's a detailed, step-by-step breakdown:
1. Step 6 Simplification:
[tex]\[
\frac{b^2 - 4ac}{4a^2} = \left(x + \frac{b}{2a}\right)^2
\][/tex]
This step ensures that the quadratic expression is simplified into a perfect square on one side and a fraction on the other side.
2. Step 7 Square Root:
[tex]\[
\sqrt{\frac{b^2 - 4ac}{4a^2}} = \sqrt{\left(x + \frac{b}{2a}\right)^2}
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
\frac{\pm\sqrt{b^2 - 4ac}}{2a} = x + \frac{b}{2a}
\][/tex]
Here, the entire expression within the square root on the left side is taken as a whole.
3. Understanding the Restriction:
In the given form ([tex]\(b^2 - 4ac\)[/tex]), separating and individually calculating the square roots of the terms ([tex]\(b^2\)[/tex] and [tex]\(-4ac\)[/tex]) before combining their results is not mathematically correct. This is because the square root operation distributes over multiplication and division but not addition and subtraction. Hence:
[tex]\[
\sqrt{b^2 - 4ac} \neq \sqrt{b^2} - \sqrt{4ac}
\][/tex]
These steps explain why the original expression [tex]\( \pm \sqrt{b^2 - 4ac} \)[/tex] must remain whole and cannot be split into [tex]\( D \pm \sqrt{-4ac} \)[/tex]. Therefore, the correct explanation is that the square root of terms separated by addition and subtraction cannot be calculated individually.
