{"id":"cb_practice_pdf:t7-math-m2-q27","source_id":"cb_practice_pdf","source_item_id":"t7-math-m2-q27","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"spr","answer_source":"rationale","layout_warning":"Part of this question was typeset outside the text flow — a fraction, subscript or similar — and PDF extraction re-inserted it in the wrong place. The wording is reliable; the symbols may be out of order.","form_id":"cb_practice_pdf:test-7","module_number":2,"position":27,"stem":"= −2.5 y = +8x+k y x^2 In the given system of equations, k is a positive integer constant. The system has no real solutions. What is the least possible value of k? OP","stimulus":"","stimulus_html":null,"stem_html":"<p>= −2.5 y = +8x+k y x^2 In the given system of equations, k is a positive integer constant. The system has no real solutions. What is the least possible value of k? OP</p>","rationale_html":"<p>14 The correct answer is . It’s given by the first equation of the system of =-2.5 -2.5 equations that y . Substituting for y in the second given equation, = 2+8 + -2.5= 2+8 + 2.5 y x x k, yields x x k. Adding to both sides of this 0= 2+8 + +2.5 equation yields x x k . A quadratic equation of the form 0= 2+ + ax bx c, where a, b, and c are constants, has no real solutions if and only 2-4 0= 2+8 + +2.5 if its discriminant, b ac, is negative. In the equation x x k , = = = +2.5 where k is a positive integer constant, a 1, b 8, and c k . Substituting 1 +2.5 2-4 82-4 1 +2.5 for a, 8 for b, and k for c in b ac yields ^ h^k h, or 64-4 +2.5 64-4 +2.5 &lt;0 ^k h. Since this value must be negative, ^k h . Adding 4 +2.5 64&lt;4 +2.5 ^k h to both sides of this inequality yields ^k h. Dividing both 16&lt; +2.5 2.5 sides of this inequality by 4 yields k . Subtracting from both sides of 13.5&lt; this inequality yields k. Since k is a positive integer constant, the least 14 possible value of k is . 46 SAT PRACTICE TEST #7 ANSWER EXPLANATIONS</p>","correct_answer":["14"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}