{"id":"cb_question_bank:551a512c-9703-43b1-b8c0-fd0af4e89ba9","source_id":"cb_question_bank","source_item_id":"551a512c-9703-43b1-b8c0-fd0af4e89ba9","external_id":"c9d1415e-a7ee-4cc6-826b-fc650c58fc3b","ibn":null,"question_id":"ebb717ab","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":"x squared minus 34 x plus c equals 0 In the given equation, c is a constant. The equation has no real solutions if c greater than n . What is the least possible value of n ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"x squared minus 34 x plus c equals 0\"><mrow>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>-</mo>\n\t\t<mrow>\n\t\t\t<mn>34</mn>\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></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"c\"><mi>c</mi>\n</math> is a constant. The equation has no real solutions if <math alttext=\"c greater than n\"><mi>c</mi><mo>&gt;</mo><mi>n</mi></math>. What is the least possible value of <math alttext=\"n\"><mi>n</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"289\"><mn>289</mn>\n</math>. A quadratic equation of the form <math alttext=\"a x squared plus b x plus c equals 0\"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></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 solutions when the value of the discriminant, <math alttext=\"b squared minus 4 a c\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mi>c</mi></math>, is less than <math alttext=\"0\"><mn>0</mn>\n</math>. In the given equation, <math alttext=\"x squared minus 34 x plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>34</mn><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>, <math alttext=\"a equals 1\"><mi>a</mi><mo>=</mo><mn>1</mn></math> and <math alttext=\"b equals negative 34\"><mi>b</mi><mo>=</mo><mo>-</mo><mn>34</mn></math>. Therefore, the discriminant of the given equation can be expressed as <math alttext=\"left parenthesis negative 34 right parenthesis squared minus 4 left parenthesis 1 right parenthesis left parenthesis c right parenthesis\"><msup><mfenced><mrow><mo>-</mo><mn>34</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mi>c</mi></mfenced></math>, or <math alttext=\"1,156 minus 4 c\"><mn>1,156</mn><mo>-</mo><mn>4</mn><mi>c</mi></math>. It follows that the given equation has no real solutions when <math alttext=\"1,156 minus 4 c less than 0\"><mn>1,156</mn><mo>-</mo><mn>4</mn><mi>c</mi><mo>&lt;</mo><mn>0</mn></math>. Adding <math alttext=\"4 c\"><mrow>\n\t<mn>4</mn>\n\t<mi>c</mi>\n</mrow>\n</math> to both sides of this inequality yields <math alttext=\"1,156 less than 4 c\"><mn>1,156</mn><mo>&lt;</mo><mn>4</mn><mi>c</mi></math>. Dividing both sides of this inequality by <math alttext=\"4\"><mn>4</mn>\n</math> yields <math alttext=\"289 less than c\"><mn>289</mn><mo>&lt;</mo><mi>c</mi></math>, or <math alttext=\"c greater than 289\"><mi>c</mi><mo>&gt;</mo><mn>289</mn></math>. It's given that the equation&nbsp;<math alttext=\"x squared minus 34 x plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>34</mn><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> has no real solutions when <math alttext=\"c greater than n\"><mi>c</mi><mo>&gt;</mo><mi>n</mi></math>. Therefore, the least possible value of <math alttext=\"n\"><mi>n</mi>\n</math> is <math alttext=\"289\"><mn>289</mn>\n</math>.</p>","correct_answer":["289"],"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}}