{"id":"cb_practice_pdf:t4-math-m2-q21","source_id":"cb_practice_pdf","source_item_id":"t4-math-m2-q21","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":null,"form_id":"cb_practice_pdf:test-4","module_number":2,"position":21,"stem":"−x^2 + bx−67 6 = 0 In the given equation, b is a positive integer. The equation has no real solution. What is the greatest possible value of b?","stimulus":"","stimulus_html":null,"stem_html":"<p>−x^2 + bx−67 6 = 0 In the given equation, b is a positive integer. The equation has no real solution. What is the greatest possible value of b?</p>","rationale_html":"<p>2 + + = The correct answer is 51. A quadratic equation of the form ax bx c 0, where a, b, and c are constants, has no real solution if and only if its discriminant, - 2+ =- =- 4ac b , is negative. In the given equation, a 1 and c 676. Substituting - - 1 for a and 676 for c in this expression yields a discriminant of 2 -( -)( - ) 2- b 4 1 676 , or b 2,704. Since this value must be negative, 2 - &lt; 2&lt; b 2,704 0, or b 2,704. Taking the positive square root of each side of this &lt; inequality yields b 52. Since b is a positive integer, and the greatest integer less than 52 is 51, the greatest possible value of b is 51.</p>","correct_answer":["51"],"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}}