{"id":"cb_question_bank:4ca746b7-f356-4bf4-90dc-0ed4c1a113e9","source_id":"cb_question_bank","source_item_id":"4ca746b7-f356-4bf4-90dc-0ed4c1a113e9","external_id":"d92caabb-d558-4de0-984a-c30db0f20ddd","ibn":null,"question_id":"13e57f0a","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":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"minus 4 x squared minus 7 x equals negative 36 What is the positive solution to the given equation?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t\n\t\t\t-\n\t\t\t4\n\t\t\t\n\t\t\t\tx\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t-\n\t\t\n\t\t\t7\n\t\t\tx\n\t\t\n\t\n\t=\n\t-36\n\n

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What is the positive solution to the given equation?

","rationale_html":"

Choice B is correct. Multiplying each side of the given equation by -16\n yields \n\t\n\t\t\n\t\t\t64\n\t\t\t\n\t\t\t\tx\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t\n\t\t\t112\n\t\t\tx\n\t\t\n\t\n\t=\n\t576\n\n. To complete the square, adding 49\n to each side of this equation yields 64x2+112x+49=576+49, or 8x+72=625. Taking the square root of each side of this equation yields two equations: \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t7\n\t\n\t=\n\t25\n\n and \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t7\n\t\n\t=\n\t-25\n\n. Subtracting 7\n from each side of the equation \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t7\n\t\n\t=\n\t25\n\n yields \n\t\n\t\t8\n\t\tx\n\t\n\t=\n\t18\n\n. Dividing each side of this equation by 8\n yields x=188, or \n\tx\n\t=\n\t\n\t\t9\n\t\t4\n\t\n\n. Therefore, \n\t9\n\t4\n\n is a solution to the given equation. Subtracting 7\n from each side of the equation \n\t\n\t\t\n\t\t\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\t7\n\t\n\t=\n\t-25\n\n yields \n\t\n\t\t8\n\t\tx\n\t\n\t=\n\t-32\n\n. Dividing each side of this equation by 8\n yields \n\tx\n\t=\n\t-4\n\n. Therefore, the given equation has two solutions, \n\t9\n\t4\n\n and -4\n. Since \n\t9\n\t4\n\n is positive, it follows that \n\t9\n\t4\n\n is the positive solution to the given equation.

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Alternate approach: Adding \n\t4\n\t\n\t\tx\n\t\t2\n\t\n\n and \n\t7\n\tx\n\n to each side of the given equation yields \n\t0\n\t=\n\t\n\t\t\n\t\t\t4\n\t\t\t\n\t\t\t\tx\n\t\t\t\t2\n\t\t\t\n\t\t\n\t\t+\n\t\t\n\t\t\t7\n\t\t\tx\n\t\t\n\t\t-\n\t\t36\n\t\n\n. The right-hand side of this equation can be rewritten as 4x2+16x-9x-36. Factoring out the common factor of \n\t4\n\tx\n\n from the first two terms of this expression and the common factor of -9\n from the second two terms yields 4xx+4-9x+4. Factoring out the common factor of x+4 from these two terms yields the expression 4x-9x+4. Since this expression is equal to 0\n, it follows that either \n\t\n\t\t\n\t\t\t4\n\t\t\tx\n\t\t\n\t\t-\n\t\t9\n\t\n\t=\n\t0\n\n or \n\t\n\t\tx\n\t\t+\n\t\t4\n\t\n\t=\n\t0\n\n. Adding 9\n to each side of the equation \n\t\n\t\t\n\t\t\t4\n\t\t\tx\n\t\t\n\t\t-\n\t\t9\n\t\n\t=\n\t0\n\n yields \n\t\n\t\t4\n\t\tx\n\t\n\t=\n\t9\n\n. Dividing each side of this equation by 4\n yields \n\tx\n\t=\n\t\n\t\t9\n\t\t4\n\t\n\n. Therefore, \n\t9\n\t4\n\n is a positive solution to the given equation. Subtracting 4\n from each side of the equation \n\t\n\t\tx\n\t\t+\n\t\t4\n\t\n\t=\n\t0\n\n yields \n\tx\n\t=\n\t-4\n\n. Therefore, the given equation has two solutions, \n\t9\n\t4\n\n and -4\n. Since \n\t9\n\t4\n\n is positive, it follows that \n\t9\n\t4\n\n is the positive solution to the given equation.

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Choice A is incorrect. Substituting \n\t7\n\t4\n\n for x\n in the given equation yields -492=-36, which is false.

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Choice C is incorrect. Substituting 4\n for x\n in the given equation yields -92=-36, which is false.

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Choice D is incorrect. Substituting 7\n for x\n in the given equation yields -245=-36, which is false.

","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"

\n\t7\n\t4\n\n

","ord":0},{"label":"B","content_html":"

\n\t9\n\t4\n\n

","ord":1},{"label":"C","content_html":"

4\n

","ord":2},{"label":"D","content_html":"

7\n

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