{"id":"cb_question_bank:7c5ae642-edcb-4b70-b978-a8fbb6404640","source_id":"cb_question_bank","source_item_id":"7c5ae642-edcb-4b70-b978-a8fbb6404640","external_id":"30e5c8c1-d494-443c-a28f-3d8867feefed","ibn":null,"question_id":"158591f0","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"x left parenthesis x plus 1 right parenthesis minus 56 equals 4 x left parenthesis x minus 7 right parenthesis What is the sum of the solutions to the given equation?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t\n\t\t\tx\n\t\t\t\n\t\t\t\t\n\t\t\t\t\tx\n\t\t\t\t\t+\n\t\t\t\t\t1\n\t\t\t\t\n\t\t\t\n\t\t\n\t\t-\n\t\t56\n\t\n\t=\n\t\n\t\t4\n\t\tx\n\t\t\n\t\t\t\n\t\t\t\tx\n\t\t\t\t-\n\t\t\t\t7\n\t\t\t\n\t\t\n\t\n\n

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What is the sum of the solutions to the given equation?

","rationale_html":"

The correct answer is 293. Applying the distributive property to the left-hand side of the given equation, xx+1-56, yields x2+x-56. Applying the distributive property to the right-hand side of the given equation, 4xx-7, yields 4x2-28x. Thus, the equation becomes x2+x-56=4x2-28x. Combining like terms on the left- and right-hand sides of this equation yields 0=4x2-x2+-28x-x+56, or 3x2-29x+56=0. For a quadratic equation in the form ax2+bx+c=0, where a\n, b\n, and c\n are constants, the quadratic formula gives the solutions to the equation in the form x=-b±b2-4ac2a. Substituting 3\n for a\n, -29\n for b\n, and 56\n for c\n from the equation 3x2-29x+56=0 into the quadratic formula yields x=29±-292-435623, or x=296±136. It follows that the solutions to the given equation are 296+136 and 296-136. Adding these two solutions gives the sum of the solutions: 296+136+296-136, which is equivalent to 296+296, or 293. Note that 29/3, 9.666, and 9.667 are examples of ways to enter a correct answer.

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