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Advanced Math / Equivalent expressions Difficulty: Medium

If $x squared equals, a, plus b$ and $y squared equals, a, plus c$ , which of the following is equal to $open parenthesis, x squared, minus y squared, close parenthesis, squared$ ?

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Explanation

Choice B is correct. It’s given that $x squared equals, a, plus b$ and $y squared equals, a, plus c$ . Using the distributive property, the expression $open parenthesis, x squared, minus y squared, close parenthesis, squared$ can be rewritten as $open parenthesis, x squared, close parenthesis, squared, minus 2 x squared, times y squared, plus, open parenthesis, y squared, close parenthesis, squared$ . Substituting $a, plus b$ and $a, plus c$ for $x squared$ and $y squared$ , respectively, in this expression yields $open parenthesis, a, plus b, close parenthesis, squared, minus, 2 times, open parenthesis, open parenthesis, a, plus b, close parenthesis, times, open parenthesis, a, plus c, close parenthesis, close parenthesis, plus, open parenthesis, a, plus c, close parenthesis, squared$ . Expanding this expression yields $open parenthesis, a, squared, plus 2 a, b, plus b squared, close parenthesis, minus, open parenthesis, 2 a, squared, plus 2 b c, plus 2 a, c, plus 2 a, b, close parenthesis, plus, open parenthesis, a, squared, plus 2 a, c, plus c squared, close parenthesis$ . Combining like terms, this expression can be rewritten as $b squared, minus 2 b c, plus c squared$ .

Choices A, C, and D are incorrect and may result from an error in using the distributive property, substituting, or combining like terms.