Column A: \((1+\sqrt{2})^2\)
Column B: 3
Column B: 3
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 A
Solution and Explanation
This question requires expanding a binomial squared and then comparing the result to an integer.
Step 2: Key Formula or Approach:
We use the formula for a perfect square: \((x+y)^2 = x^2 + 2xy + y^2\).
Step 3: Detailed Explanation:
Let's expand the expression in Column A.
\[ (1+\sqrt{2})^2 = (1)^2 + 2(1)(\sqrt{2}) + (\sqrt{2})^2 \] \[ = 1 + 2\sqrt{2} + 2 \] \[ = 3 + 2\sqrt{2} \] Step 4: Comparing the Quantities:
Now we compare Column A with Column B.
Column A: \(3 + 2\sqrt{2}\)
Column B: 3
Since \(\sqrt{2}\) is a positive number (approximately 1.414), \(2\sqrt{2}\) is also a positive number.
Therefore, the value in Column A is 3 plus a positive amount, which is strictly greater than 3.
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