{"id":"cb_question_bank:d7e96295-34ee-4903-a32b-4abce70f60af","source_id":"cb_question_bank","source_item_id":"d7e96295-34ee-4903-a32b-4abce70f60af","external_id":"00272e26-db4e-4b57-b229-9040d402e42f","ibn":null,"question_id":"9cb9beec","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":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"y equals negative 1.50 y equals x squared plus 8 x plus a In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?","stimulus":"","stimulus_html":null,"stem_html":"

\n\ty\n\t=\n\t-1.50\n\n

\n

\n\ty\n\t=\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\t8\n\t\t\tx\n\t\t\n\t\t+\n\t\ta\n\t\n\n

\n

In the given system of equations, a\n is a positive constant. The system has exactly one distinct real solution. What is the value of a\n?

","rationale_html":"

The correct answer is 292. According to the first equation in the given system, the value of y\n is -1.5\n. Substituting -1.5\n for y\n in the second equation in the given system yields -1.5=x2+8x+a. Adding 1.5\n to both sides of this equation yields 0=x2+8x+a+1.5. If the given system has exactly one distinct real solution, it follows that 0=x2+8x+a+1.5 has exactly one distinct real solution. A quadratic equation in the form 0=px2+qx+r, where p\n, q\n, and r\n are constants, has exactly one distinct real solution if and only if the discriminant, q2-4pr, is equal to 0\n. The equation 0=x2+8x+a+1.5 is in this form, where p=1, q=8, and r=a+1.5. Therefore, the discriminant of the equation 0=x2+8x+a+1.5 is 82-41a+1.5, or 58-4a. Setting the discriminant equal to 0\n to solve for a\n yields 58-4a=0. Adding \n\t4\n\ta\n\n to both sides of this equation yields 58=4a. Dividing both sides of this equation by 4\n yields 584=a, or 292=a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a\n is 292. Note that 29/2 and 14.5 are examples of ways to enter a correct answer.

","correct_answer":["14.5","29/2"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.823000+00:00","source_updated_at":"2023-08-02T20:25:59.823000+00:00","options":[],"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}}