Mathematics > Standard Multiple Choice
Read the following SAT test question and then click on a button to select your answer.
If (x + y)^2 = (x^2) + (y^2), which of the following statements must also be true?
Roman numeral 1. x = 0
Roman numeral 2.(x minus y)^2 = (x^2) + (y^2)
Roman numeral 3. x times y = 0
(A) None
(B) Roman numeral 1 only
(C) Roman numeral 2 only
(D) Roman numeral 3 only
(E) Roman numeral 2 and Roman numeral 3
The correct answer is E
Explanation
The quantity (x + y)^2 can be expressed as (x^2) + (2 times x times y) + (y^2). If (x + y)^2 = (x^2) + (y^2), then 2 times x times y = 0 and x times y = 0. Since x times y = 0, either x = 0 or y = 0 or both. Therefore, statement Roman numeral 3 must be true, but statement Roman numeral 1, x = 0, is not always true. For statement Roman numeral 2, you can write (x minus y)^2 = (x^2) minus (2 times x times y) + y^2, and since x times y = 0, it follows that (x minus y)^2 = (x^2) + (y^2). Therefore, both statements Roman numeral 2 and Roman numeral 3 must be true.