If x = √3 + 1 and y = √3 − 1, then what is x<sup>2</sup> + y<sup>2</sup>?
Aptitude
Square Root and Cube Root
Difficulty: Easy
Choose an option
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A6
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B7
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C8
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D10
Answer
Correct Answer: 8
Explanation
Problem restatementEvaluate x2 + y2 for x = √3 + 1 and y = √3 − 1.
Given data
- x = √3 + 1
- y = √3 − 1
Concept/ApproachSquare each expression and add; the radical cross-terms cancel.
Step-by-step calculation x2 = (√3 + 1)2 = 3 + 2√3 + 1 = 4 + 2√3 y2 = (√3 − 1)2 = 3 − 2√3 + 1 = 4 − 2√3 x2 + y2 = (4 + 2√3) + (4 − 2√3) = 8
Verification/AlternativeUse identity: (a + b)2 + (a − b)2 = 2(a2 + b2) with a = √3, b = 1 ⇒ 2(3 + 1) = 8.
Final Answer8