{"id":"cb_question_bank:3e002877-af71-4528-b329-5f1152832fe9","source_id":"cb_question_bank","source_item_id":"3e002877-af71-4528-b329-5f1152832fe9","external_id":"cc8d5c5d-bc43-4311-981f-7eb0259aaf7a","ibn":null,"question_id":"1fe32f7d","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":"minus x squared plus b x minus 676 equals 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 style=\"text-align: center;\"><math alttext=\"minus x squared plus b x minus 676 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<msup>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mi>b</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>676</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math></p>\n<p>In the given equation, <math alttext=\"b\"><mi>b</mi>\n</math> is a positive integer. The equation has no real solution. What is the greatest possible value of <math alttext=\"b\"><mi>b</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"51\"><mn>51</mn>\n</math>. A quadratic equation of the form <math alttext=\"a x squared plus b x plus c equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mi>a</mi>\n\t\t\t<msup>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mi>b</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mi>c</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants, has no real solution if and only if its discriminant, <math alttext=\"minus 4 a c plus b squared\"><mrow>\n\t<mrow>\n\t\t<mo>-</mo>\n\t\t<mn>4</mn>\n\t\t<mi>a</mi>\n\t\t<mi>c</mi>\n\t</mrow>\n\t<mo>+</mo>\n\t<msup>\n\t\t<mi>b</mi>\n\t\t<mn>2</mn>\n\t</msup>\n</mrow>\n</math>, is negative. In the given equation, <math alttext=\"a equals negative 1\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>1</mn>\n</mrow>\n</math> and <math alttext=\"c equals negative 676\"><mrow>\n\t<mi>c</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>676</mn>\n</mrow>\n</math>. Substituting <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"negative 676\"><mo>-</mo><mn>676</mn>\n</math> for <math alttext=\"c\"><mi>c</mi>\n</math> in this expression yields a discriminant of <math alttext=\"b squared minus 4 left parenthesis negative 1 right parenthesis left parenthesis negative 676 right parenthesis\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>676</mn></mrow></mfenced></math>, or <math alttext=\"b squared minus 2,704\"><mrow>\n\t<msup>\n\t\t<mi>b</mi>\n\t\t<mn>2</mn>\n\t</msup>\n\t<mo>-</mo>\n\t<mn>2,704</mn>\n</mrow>\n</math>. Since this value must be negative,&nbsp;<math alttext=\"b squared minus 2,704 less than 0\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2,704</mn><mo>&lt;</mo><mn>0</mn></math>, or <math alttext=\"b squared less than 2,704\"><msup><mi>b</mi><mn>2</mn></msup><mo>&lt;</mo><mn>2,704</mn></math>. Taking the positive square root of each side of this inequality yields&nbsp;<math alttext=\"b less than 52\"><mi>b</mi><mo>&lt;</mo><mn>52</mn></math>. Since <math alttext=\"b\"><mi>b</mi>\n</math> is a positive integer, and the greatest integer less than <math alttext=\"52\"><mn>52</mn>\n</math> is <math alttext=\"51\"><mn>51</mn>\n</math>, the greatest possible value of <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"51\"><mn>51</mn>\n</math>.</p>","correct_answer":["51"],"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":[],"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}}