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Equivalent expressions SAT question

Math · Advanced Math · Medium · Answer it here, explanation included.

The sum of $negative 2 x squared, plus x, plus 31$ and $3 x squared, plus 7 x, minus 8$ can be written in the form $a, x squared, plus b x, plus c$ , where a, b, and c are constants. What is the value of $a, plus b plus c$ ?

Enter a number or fraction (like 3/17 or 0.176). Press Enter to check.

ExplanationShow

The correct answer is 32. The sum of the given expressions is $open parenthesis, negative 2, x squared, plus x, plus 31, close parenthesis, plus, open parenthesis, 3 x squared, plus 7 x, minus 8, close parenthesis$ . Combining like terms yields $x squared, plus 8 x, plus 23$ . Based on the form of the given equation, $a, equals 1$ , $b equals 8$ , and $c equals 23$ . Therefore, $a, plus b, plus c, equals 32$ .

Alternate approach: Because $a, plus b, plus c$ is the value of $a, x squared, plus b x, plus c$ when $x equals 1$ , it is possible to first make that substitution into each polynomial before adding them. When $x equals 1$ , the first polynomial is equal to $negative 2 plus 1, plus 31, equals 30$ and the second polynomial is equal to $3 plus 7, minus 8, equals 2$ . The sum of 30 and 2 is 32.

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