Question

\((x - 2y)(x + 2y) = 4\)
Column A: \(x^2 - 4y^2\)
Column B: 8

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

Always be on the lookout for common algebraic identities like the difference of squares (\(a^2 - b^2\)), perfect squares (\((a+b)^2, (a-b)^2\)), and sum/difference of cubes. Recognizing them can simplify a problem instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests the knowledge of a fundamental algebraic identity, the "difference of squares".
Step 2: Key Formula or Approach:
The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).
Step 3: Detailed Explanation:
We need to evaluate the expression in Column A, which is \(x^2 - 4y^2\).
We can rewrite \(4y^2\) as \((2y)^2\). So the expression becomes \(x^2 - (2y)^2\).
This expression is in the form \(a^2 - b^2\), where \(a = x\) and \(b = 2y\).
Applying the difference of squares formula, we get:
\[ x^2 - (2y)^2 = (x - 2y)(x + 2y) \] The problem gives us the information that \((x - 2y)(x + 2y) = 4\).
Therefore, the value of the expression in Column A is exactly 4.
\[ x^2 - 4y^2 = 4 \] Step 4: Comparing the Quantities:
Column A: \(x^2 - 4y^2 = 4\)
Column B: 8
Comparing the two values, we see that \(4 \textless 8\).
Thus, the quantity in Column B is greater.