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Advanced Math / Equivalent expressions Difficulty: Medium

$open parenthesis, one half x, plus three halves, close parenthesis, times, open parenthesis, three halves x, plus one half, close parenthesis$

The expression above is equivalent to $a, x squared, plus b x, plus c$ , where a, b, and c are constants. What is the value of b?

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Explanation

The correct answer is $five halves$ . The expression $open parenthesis, one half x, plus three halves, close parenthesis, times, open parenthesis, three halves x, plus one half, close parenthesis$ can be written in the form $a, x squared, plus b x, plus c$ , where a, b, and c are constants, by multiplying out the expression using the distributive property of multiplication over addition. The result is $one half x, times three halves x, plus, one half x, times one half, plus, three halves times three halves x, plus, three halves times one half$ . This expression can be rewritten by multiplying as indicated to give $three fourths x squared, plus, one fourth x, plus, nine fourths x, plus, three fourths$ , which can be simplified to $three fourths x squared, plus, ten fourths x, plus, three fourths$ , or $three fourths x squared, plus, five halves x, plus, three fourths$ . This is in the form $a, x squared, plus b x, plus c$ , where the value of b is $five halves$ . Note that 5/2 and 2.5 are examples of ways to enter a correct answer.