\((x - 2y)(x + 2y) = 4\)
Column A: \(x^2 - 4y^2\)
Column B: 8
Column A: \(x^2 - 4y^2\)
Column B: 8
Show Hint
- Quantity A is greater
- Quantity B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
The Correct Option is B
Solution and Explanation
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.
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