Answer :
To answer this question, let's consider the mathematical expression involved:
We have the discriminant of a quadratic equation, which is given by [tex]\( b^2 - 4ac \)[/tex]. This discriminant is part of the quadratic formula [tex]\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)[/tex], which is used to find the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
We need to explain why the expression [tex]\( \pm \sqrt{b^2 - 4ac} \)[/tex] cannot be rewritten as [tex]\( b \pm \sqrt{-4ac} \)[/tex].
1. Negative values, like [tex]\(-4ac\)[/tex], do not have a square root:
- This statement is not accurate because negative values can have square roots, but in the context of real numbers, the square root of a negative number is not a real number (it becomes imaginary). Also, in this specific problem context, the issue is not about negative values per se, but the structure of the discriminant.
2. The [tex]\(\pm\)[/tex] symbol prevents the square root from being evaluated:
- This statement is somewhat misleading. The [tex]\(\pm\)[/tex] symbol simply indicates that we consider both the positive and negative roots when solving the quadratic equation. It does not inherently prevent evaluating the square root.
3. The square root of terms separated by addition and subtraction cannot be calculated individually:
- This is the correct explanation. The discriminant [tex]\( b^2 - 4ac \)[/tex] is a single term under the square root sign. The entire discriminant [tex]\( b^2 - 4ac \)[/tex] must be considered together before taking the square root. We cannot individually take the square root of [tex]\( b^2 \)[/tex] and [tex]\( -4ac \)[/tex] and then simplify the expression.
4. The entire term [tex]\( b^2 - 4ac \)[/tex] must be divided by [tex]\(2a\)[/tex] before its square root can be determined:
- This statement is incorrect. The division by [tex]\(2a\)[/tex] happens after taking care of the square root of the discriminant. The discriminant [tex]\( b^2 - 4ac \)[/tex] is computed first, followed by the square root operation, and then division by [tex]\(2a\)[/tex].
Therefore, the best explanation is:
The square root of terms separated by addition and subtraction cannot be calculated individually.
We have the discriminant of a quadratic equation, which is given by [tex]\( b^2 - 4ac \)[/tex]. This discriminant is part of the quadratic formula [tex]\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)[/tex], which is used to find the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
We need to explain why the expression [tex]\( \pm \sqrt{b^2 - 4ac} \)[/tex] cannot be rewritten as [tex]\( b \pm \sqrt{-4ac} \)[/tex].
1. Negative values, like [tex]\(-4ac\)[/tex], do not have a square root:
- This statement is not accurate because negative values can have square roots, but in the context of real numbers, the square root of a negative number is not a real number (it becomes imaginary). Also, in this specific problem context, the issue is not about negative values per se, but the structure of the discriminant.
2. The [tex]\(\pm\)[/tex] symbol prevents the square root from being evaluated:
- This statement is somewhat misleading. The [tex]\(\pm\)[/tex] symbol simply indicates that we consider both the positive and negative roots when solving the quadratic equation. It does not inherently prevent evaluating the square root.
3. The square root of terms separated by addition and subtraction cannot be calculated individually:
- This is the correct explanation. The discriminant [tex]\( b^2 - 4ac \)[/tex] is a single term under the square root sign. The entire discriminant [tex]\( b^2 - 4ac \)[/tex] must be considered together before taking the square root. We cannot individually take the square root of [tex]\( b^2 \)[/tex] and [tex]\( -4ac \)[/tex] and then simplify the expression.
4. The entire term [tex]\( b^2 - 4ac \)[/tex] must be divided by [tex]\(2a\)[/tex] before its square root can be determined:
- This statement is incorrect. The division by [tex]\(2a\)[/tex] happens after taking care of the square root of the discriminant. The discriminant [tex]\( b^2 - 4ac \)[/tex] is computed first, followed by the square root operation, and then division by [tex]\(2a\)[/tex].
Therefore, the best explanation is:
The square root of terms separated by addition and subtraction cannot be calculated individually.
