Question

If \(x^2 - y^2 = 48\), then \( \frac{2}{3}(x+y)(x-y) = \)

• A. 16
• B. 72
• C. 96
• D. 32
• E. 64

Hint:

Recognizing algebraic identities like the difference of squares (\(a^2-b^2 = (a-b)(a+b)\)) is crucial for speed and accuracy. When you see one form, immediately think of the other. This question is a direct application of this identity.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem uses the "difference of squares" factorization, which is a fundamental identity in algebra.

Step 2: Key Formula or Approach:
The difference of squares formula is:
\[ x^2 - y^2 = (x+y)(x-y) \]

Step 3: Detailed Explanation:
We are given the value of \( x^2 - y^2 \).
\[ x^2 - y^2 = 48 \] We need to find the value of the expression \( \frac{2}{3}(x+y)(x-y) \).
Using the difference of squares formula, we can substitute \( (x+y)(x-y) \) with \( x^2 - y^2 \).
\[ \frac{2}{3}(x+y)(x-y) = \frac{2}{3}(x^2 - y^2) \] Now, substitute the given value of 48 into the expression.
\[ \frac{2}{3}(48) \] To calculate this, we can first divide 48 by 3 and then multiply by 2.
\[ \frac{48}{3} = 16 \] \[ \frac{2}{3}(48) = 2 \times 16 = 32 \]

Step 4: Final Answer:
The value of the expression is 32.