Which best explains why the expression $\pm \sqrt{b^2-4 a c}$ cannot be rewritten as $b \pm \sqrt{-4 a c}$ during the next step?
A. Negative values, like $-4 a c$, do not have a square root.
B. The $\pm$ symbol prevents the square root from being evaluated.
C. The square root of terms separated by addition and subtraction cannot be calculated individually.
D. The entire term $b^2-4 a c$ must be divided by $2 a$ before its square root can be determined.
Answer (2)
The expression is ± b 2 − 4 a c .
Square root of terms separated by addition and subtraction cannot be calculated individually.
Therefore, ± b 2 − 4 a c cannot be rewritten as b ± − 4 a c .
The correct answer is: The square root of terms separated by addition and subtraction cannot be calculated individually.
Explanation
Understanding the Problem We are given a step in the derivation of the quadratic formula: 2 a p m s q r t b 2 − 4 a c = x + 2 a b We need to determine why the expression ± b 2 − 4 a c cannot be rewritten as b ± − 4 a c .
Reasoning The key reason why ± b 2 − 4 a c cannot be rewritten as b ± − 4 a c is due to the properties of square roots. The square root of a sum or difference is not generally equal to the sum or difference of the square roots. In other words, x + y = x + y .
Conclusion Therefore, we cannot individually take the square root of b 2 and − 4 a c separately and add/subtract them. The correct answer is: The square root of terms separated by addition and subtraction cannot be calculated individually.
Examples
Understanding why you can't distribute a square root across addition or subtraction is crucial in many areas of math. For instance, when simplifying radical expressions or solving equations involving radicals, this principle helps avoid common mistakes. Imagine you're calculating the length of the hypotenuse of a right triangle using the Pythagorean theorem, a 2 + b 2 = c 2 . If you tried to say that a 2 + b 2 = a + b , you'd get the wrong answer for the length of the hypotenuse! This concept also applies in physics, such as when calculating the magnitude of a resultant vector from its components.
The expression ± b 2 − 4 a c cannot be rewritten as b ± − 4 a c because the square root of a difference cannot be simplified in such a manner. This is due to the properties of square roots, specifically that x − y = x − y . The correct answer is option C.
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