{"id":"cb_question_bank:0d1fd283-a5c4-401d-9862-45177888c935","source_id":"cb_question_bank","source_item_id":"0d1fd283-a5c4-401d-9862-45177888c935","external_id":"28f04aa6-9eca-4e86-bee8-86e106dc02df","ibn":null,"question_id":"ba0edc30","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":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"x squared minus 2 x minus 9 equals 0 One solution to the given equation can be written as 1 plus StartRoot k EndRoot , where k is a constant. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t\n\t\t\tx\n\t\t\t2\n\t\t\n\t\t-\n\t\t\n\t\t\t2\n\t\t\tx\n\t\t\n\t\t-\n\t\t9\n\t\n\t=\n\t0\n\n

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One solution to the given equation can be written as 1+k, where k\n is a constant. What is the value of k\n?

","rationale_html":"

Choice B is correct. Adding 9\n to each side of the given equation yields \n\t\n\t\t\n\t\t\tx\n\t\t\t2\n\t\t\n\t\t-\n\t\t\n\t\t\t2\n\t\t\tx\n\t\t\n\t\n\t=\n\t9\n\n. To complete the square, adding 1\n to each side of this equation yields x2-2x+1=9+1, or x-12=10. Taking the square root of each side of this equation yields x-1=±10. Adding 1\n to each side of this equation yields x=1±10. Since it's given that one of the solutions to the equation can be written as 1+k, the value of k\n must be 10\n.

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Alternate approach: It's given that 1+k is a solution to the given equation. It follows that x=1+k. Substituting 1+k for x\n in the given equation yields 1+k2-21+k-9=0, or 1+k1+k-21+k-9=0. Expanding the products on the left-hand side of this equation yields 1+2k+k-2-2k-9=0, or \n\t\n\t\tk\n\t\t-\n\t\t10\n\t\n\t=\n\t0\n\n. Adding 10\n to each side of this equation yields \n\tk\n\t=\n\t10\n\n.

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Choice A is incorrect and may result from conceptual or calculation errors.

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Choice C is incorrect and may result from conceptual or calculation errors.

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Choice D is incorrect and may result from conceptual or calculation errors.

","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":"

8\n

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

10\n

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

20\n

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

40\n

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}