{"id":"cb_question_bank:715cca3c-12e3-4e96-8e2c-3de141c0be29","source_id":"cb_question_bank","source_item_id":"715cca3c-12e3-4e96-8e2c-3de141c0be29","external_id":"a5f0747c-79a9-4cc0-a9eb-aa9d68a2d5e8","ibn":null,"question_id":"1ce9ffcd","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":"minus 9 x squared plus 30 x plus c equals 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?","stimulus":"","stimulus_html":null,"stem_html":"

\n\t\n\t\t\n\t\t\t-\n\t\t\t9\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\t30\n\t\t\tx\n\t\t\n\t\t+\n\t\tc\n\t\n\t=\n\t0\n\n

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In the given equation, c\n is a constant. The equation has exactly one solution. What is the value of c\n?

","rationale_html":"

Choice C is correct. It's given that the equation -9x2+30x+c=0 has exactly one solution. A quadratic equation of the form \n\t\n\t\t\n\t\t\ta\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\tb\n\t\t\tx\n\t\t\n\t\t+\n\t\tc\n\t\n\t=\n\t0\n\n has exactly one solution if and only if its discriminant, \n\t\n\t\t-\n\t\t4\n\t\ta\n\t\tc\n\t\n\t+\n\t\n\t\tb\n\t\t2\n\t\n\n, is equal to zero. It follows that for the given equation, \n\ta\n\t=\n\t-9\n\n and \n\tb\n\t=\n\t30\n\n. Substituting -9\n for a\n and 30\n for b\n into b2-4ac  yields 302-4-9c, or 900+36c. Since the discriminant must equal zero, 900+36c=0. Subtracting \n\t36\n\tc\n\n from both sides of this equation yields \n\t900\n\t=\n\t\n\t\t-\n\t\t36\n\t\tc\n\t\n\n. Dividing each side of this equation by -36\n yields \n\t-25\n\t=\n\tc\n\n. Therefore, the value of c\n is -25\n. 

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Choice A is incorrect. If the value of c\n is 3\n, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

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Choice B is incorrect. If the value of c\n is 0\n, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution.

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Choice D is incorrect. If the value of c\n is -53\n, this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.

","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.824000+00:00","source_updated_at":"2023-08-02T20:25:59.824000+00:00","options":[{"label":"A","content_html":"

3\n

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

0\n

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

-25\n

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

-53\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}}