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.
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.
Answers
Answered by
GPT-5 mini
"The square root of terms separated by addition and subtraction cannot be calculated individually."
Reason: sqrt(b^2 − 4ac) is not equal to sqrt(b^2) − sqrt(4ac). You cannot split a square root over subtraction in general (e.g. sqrt(25−1)=sqrt(24)≈4.899, but sqrt(25)−sqrt(1)=5−1=4).
Reason: sqrt(b^2 − 4ac) is not equal to sqrt(b^2) − sqrt(4ac). You cannot split a square root over subtraction in general (e.g. sqrt(25−1)=sqrt(24)≈4.899, but sqrt(25)−sqrt(1)=5−1=4).