Question

Some of the steps in the derivation of the quadratic formula are shown.

Step 4: StartFraction negative 4 a c + b squared Over 4 a EndFraction = a ( x + StartFraction b Over 2 a EndFraction) squared

Step 5: (StartFraction 1 Over a EndFraction) StartFraction b squared minus 4 a c Over 4 a EndFraction = (StartFraction 1 Over a EndFraction) a (x + StartFraction b Over 2 a EndFraction) squared

Step 6: StartFraction b squared minus 4 a c Over 4 a squared EndFraction = ( x + StartFraction b Over 2 a EndFraction) squared

Step 7: StartFraction plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFraction = x + StartFraction b Over 2 a EndFraction

Which best explains why the expression plus or minus StartRoot b squared minus 4 a c EndRoot cannot be rewritten as b plus or minus StartRoot negative 4 a c EndRoot during the next step?

Negative values, like −4ac, do not have a square root.
The ± symbol prevents the square root from being evaluated.
The square root of terms separated by addition and subtraction cannot be calculated individually.
The entire term b2 − 4ac must be divided by 2a before its square root can be determined.

Answer 1

"The square root of terms separated by addition and subtraction cannot be calculated individually."

Explanation: √(b^2 − 4ac) is not equal to √(b^2) − √(4ac) in general. (Also √(b^2)=|b|, not b.) You cannot split a square root over a sum or difference.