What is the simplest form of the expression below?
x − x + x − x + x − ⋯
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How do you know that the other 3 "solutions" are definitely wrong?
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I did not say that they are wrong. When presented with a multiple choice situation, the answer is the form present among the choices.
for x>0
2 1 ( − 4 x − 3 − 1 ) and 2 1 ( 1 − 4 x + 1 ) < 0
This will mean y<0 which is not possible as square root of a number denotes the positive root.
if y = 2 1 ( 1 + 4 x + 1 ) y 2 = x + y ⟹ y = x + x + x + ⋯
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The principal square root is the positive root. The square root operation has two roots, positive and negative.
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Solving y = x − x + y gives four solutions: 2 1 ( − 4 x − 3 − 1 ) , 2 1 ( 4 x − 3 − 1 ) , 2 1 ( 1 − 4 x + 1 ) and 2 1 ( 4 x + 1 + 1 ) . Only one of those solutions is among the multiple choices offered as a solution. Therefore, that has to be the desired answer; even though I would have selected 2 1 ( 4 x + 1 + 1 ) as the simplest form.