Is a² > b²? (1) a² + b² = 2ab (2) (a + b)² = (a – b)²

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(A) Statement (1) is a² + b² = 2ab. a² – 2ab + b² = 0 (a – b)² = 0 a = b So a² = b². We have the definite answer to the main question (NO). Therefore statement (1) by itself is sufficient.

Statement (2) is (a + b)² = (a – b)². a² + 2ab + b² = a² – 2ab + b² 4ab = 0 So either a, b or both equal 0. The other one can possess any value. Therefore we do not have a definite answer to the main question. Statement (2) by itself is NOT sufficient.

Statement (1) by itself is sufficient to answer the question, but statement (2) by itself is not. The correct answer is A. ---------- In statement (2), can we say that since (a + b)² = (a – b)², then (a + b) = (a – b) ?

This way Statement(2) will be sufficient to answer the question. Regards.

In statement (2), can we say that since (a + b)² = (a – b)², then (a + b) = (a – b) ?

No, we can not. Consider x² = y² The proper transformation is |x| = |y|, so either x = y or x = -y. For example, 3² = 3², or 3 = (-3)², (-3)² = 3², (-3)² = (-3)² ,– all fit.

By simplifying x² = y² as x = y we loose the solution x = -y and all the corresponding possible values.

The explanation to this problem shows that the expression (a + b)² = (a – b)² holds true if a = 0 and b is any number: b² = (-b)² or b = 0 and a is any number: a² = a²

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