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Advanced Math / Equivalent expressions Difficulty: Medium

Which of the following is an equivalent form of $open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, squared, minus, open parenthesis, 5 point 2, x squared, minus 6 point 4, close parenthesis$ ?

A. $negative 2 point 2, x squared, plus 1 point 6$

B. $negative 2 point 2, x squared, plus 11 point 2$

C. $negative 2 point 95, x squared, minus 7 point 2 x, plus 12 point 16$

D. $negative 2 point 95, x squared, minus 7 point 2 x, plus 0 point 64$

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Explanation

Choice C is correct. The first expression $open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, squared$ can be rewritten as $open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis, times, open parenthesis, 1 point 5 x minus 2 point 4, close parenthesis$ . Applying the distributive property to this product yields $open parenthesis, 2 point 2 5, x squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, close parenthesis, minus, open parenthesis, 5 point 2, x squared, minus 6 point 4, close parenthesis$ . This difference can be rewritten as $open parenthesis, 2 point 2 5, x squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, close parenthesis, plus, negative 1 times, open parenthesis, 5 point 2 x squared, minus 6 point 4, close parenthesis$ . Distributing the factor of $negative 1$ through the second expression yields $2 point 2 5 x, squared, minus 3 point 6 x, minus 3 point 6 x, plus 5 point 7 6, minus 5 point 2, x squared, plus 6 point 4$ . Regrouping like terms, the expression becomes $open parenthesis, 2 point 2 5, x squared, minus 5 point 2, x squared, close parenthesis, plus, open parenthesis, negative 3 point 6 x minus 3 point 6 x, close parenthesis, plus, open parenthesis, 5 point 7 6 plus 6 point 4, close parenthesis$ . Combining like terms yields $negative 2 point 9 5, x squared, minus 7 point 2 x, plus 12 point 1 6$ .

Choices A, B, and D are incorrect and likely result from errors made when applying the distributive property or combining the resulting like terms.