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Equivalent expressions SAT question

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

$2, x squared, plus 5 x, minus 12$

If the given expression is rewritten in the form $open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis$ , where k is a constant, what is the value of k ?

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

ExplanationShow

The correct answer is 4. It’s given that $2 x squared, plus 5 x, minus 12$ can be rewritten as $open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis$ ; it follows that $open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis, equals, 2 x squared, plus 5 x, minus 12$ . Expanding the left-hand side of this equation yields $2 x squared, plus 2 k x, minus 3 x, minus 3 k, equals, 2 x squared, plus 5 x, minus 12$ . Subtracting $2 x squared$ from both sides of this equation yields $2 k x minus 3 x, minus 3 k, equals, 5 x minus 12$ . Using properties of equality, $2 k x minus 3 x, equals 5 x$ and $negative 3 k, equals negative 12$ . Either equation can be solved for k. Dividing both sides of $negative 3 k, equals negative 12$ by $negative 3$ yields $k equals 4$ . The equation $2 k x minus 3 x, equals 5 x$ can be rewritten as $x times, open parenthesis, 2 k minus 3, close parenthesis, equals 5 x$ . It follows that $2 k minus 3, equals 5$ . Solving this equation for k also yields $k equals 4$ . Therefore, the value of k is 4.

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