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Advanced Math / Equivalent expressions Difficulty: Medium

Which of the following expressions is equivalent to $x squared minus 5$ ?

A. $open parenthesis, x plus the square root of 5, close parenthesis, squared$

B. $open parenthesis, x minus the square root of 5, close parenthesis, squared$

C. $open parenthesis, x plus the square root of 5, close parenthesis, times, open parenthesis, x minus the square root of 5, close parenthesis$

D. $open parenthesis, x plus 5, close parenthesis, times, open parenthesis, x minus 1, close parenthesis$

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Explanation

Choice C is correct. The expression can be written as a difference of squares x² – y², which can be factored as (x + y)(x – y). Here, y² = 5, so $y equals the square root of 5$ , and the expression therefore factors as $open parenthesis, x plus the square root of 5, close parenthesis, times, open parenthesis, x minus the square root of 5, close parenthesis$ .

Choices A and B are incorrect and may result from misunderstanding how to factor a difference of squares. Choice D is incorrect; (x + 5)(x – 1) can be rewritten as x² + 4x – 5, which is not equivalent to the original expression.