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Advanced Math / Equivalent expressions Difficulty: Hard

$the fraction with numerator x squared, minus c, and denominator x minus b, end fraction$

In the expression above, b and c are positive integers. If the expression is equivalent to $x plus b$ and $x is not equal to b$ , which of the following could be the value of c ?

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Explanation

Choice A is correct. If the given expression is equivalent to $x plus b$ , then $the fraction with numerator x squared, minus c, and denominator x minus b, end fraction, equals, x plus b$ , where x isn’t equal to b. Multiplying both sides of this equation by $x minus b$ yields $x squared, minus c, equals, open parenthesis, x plus b, close parenthesis, times, open parenthesis, x minus b, close parenthesis$ . Since the right-hand side of this equation is in factored form for the difference of squares, the value of c must be a perfect square. Only choice A gives a perfect square for the value of c.

Choices B, C, and D are incorrect. None of these values of c produces a difference of squares. For example, when 6 is substituted for c in the given expression, the result is $the fraction with numerator x squared, minus 6, and denominator x minus b, end fraction$ . The expression $x squared, minus 6$ can’t be factored with integer values, and therefore $the fraction with numerator x squared, minus 6, and denominator x minus b, end fraction$ isn’t equivalent to $x plus b$ .