{"id":"cb_question_bank:4b568321-a580-4bb9-96ea-860cc9d1e52a","source_id":"cb_question_bank","source_item_id":"4b568321-a580-4bb9-96ea-860cc9d1e52a","external_id":"75273fff-e1a3-4a75-a92d-7cecf27c2bd7","ibn":null,"question_id":"03ff48d2","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 left parenthesis k x minus 56 right parenthesis equals negative 16 In the given equation, k is an integer constant. If the equation has no real solution, what is the least possible value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"x left parenthesis k x minus 56 right parenthesis equals negative 16\"><mrow>\n\t<mrow>\n\t\t<mi>x</mi>\n\t\t<mfenced>\n\t\t\t<mrow>\n\t\t\t\t<mrow>\n\t\t\t\t\t<mi>k</mi>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t</mrow>\n\t\t\t\t<mo>-</mo>\n\t\t\t\t<mn>56</mn>\n\t\t\t</mrow>\n\t\t</mfenced>\n\t</mrow>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>16</mn>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"k\"><mi>k</mi>\n</math> is an integer constant. If the equation has no real solution, what is the least possible value of <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"50\"><mn>50</mn>\n</math>. An equation of the form&nbsp;<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&nbsp;no real solutions if and only if its discriminant,&nbsp;<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 negative. Applying the distributive property to the left-hand side of the equation <math alttext=\"x left parenthesis k x minus 56 right parenthesis equals negative 16\"><mi>x</mi><mfenced><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>56</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>16</mn></math> yields <math alttext=\"k x squared minus 56 x equals negative 16\"><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>56</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>16</mn></math>. Adding <math alttext=\"16\"><mn>16</mn>\n</math> to each side of this equation yields <math alttext=\"k x squared minus 56 x plus 16 equals 0\"><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>56</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></math>. Substituting <math alttext=\"k\"><mi>k</mi>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"negative 56\"><mo>-</mo><mn>56</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"16\"><mn>16</mn>\n</math> for <math alttext=\"c\"><mi>c</mi>\n</math> in <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> yields a discriminant of&nbsp;<math alttext=\"left parenthesis negative 56 right parenthesis squared minus 4 left parenthesis k right parenthesis left parenthesis 16 right parenthesis\"><msup><mfenced><mrow><mo>-</mo><mn>56</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mi>k</mi></mfenced><mfenced><mn>16</mn></mfenced></math>, or <math alttext=\"3,136 minus 64 k\"><mn>3,136</mn><mo>-</mo><mn>64</mn><mi>k</mi></math>. If the given equation has no real solution, it follows that the value of <math alttext=\"3,136 minus 64 k\"><mn>3,136</mn><mo>-</mo><mn>64</mn><mi>k</mi></math> must be negative. Therefore, <math alttext=\"3,136 minus 64 k less than 0\"><mn>3,136</mn><mo>-</mo><mn>64</mn><mi>k</mi><mo>&lt;</mo><mn>0</mn></math>. Adding <math alttext=\"64 k\"><mrow>\n\t<mn>64</mn>\n\t<mi>k</mi>\n</mrow>\n</math> to both sides of this inequality yields <math alttext=\"3,136 less than 64 k\"><mn>3,136</mn><mo>&lt;</mo><mn>64</mn><mi>k</mi></math>. Dividing both sides of this inequality by <math alttext=\"64\"><mn>64</mn>\n</math> yields <math alttext=\"49 less than k\"><mn>49</mn><mo>&lt;</mo><mi>k</mi></math>, or <math alttext=\"k greater than 49\"><mi>k</mi><mo>&gt;</mo><mn>49</mn></math>. Since it's given that <math alttext=\"k\"><mi>k</mi>\n</math> is an integer, the least possible value of <math alttext=\"k\"><mi>k</mi>\n</math> is <math alttext=\"50\"><mn>50</mn>\n</math>.&nbsp;</p>","correct_answer":["50"],"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}}