{"id":"cb_question_bank:5cc23fde-d442-49ba-a886-4efe29c7589a","source_id":"cb_question_bank","source_item_id":"5cc23fde-d442-49ba-a886-4efe29c7589a","external_id":"22d60924-a9af-42ee-8e63-30a79298bfb3","ibn":null,"question_id":"3d12b1e0","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 16 x squared minus 8 x plus c equals 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"minus 16 x squared minus 8 x plus c equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>16</mn>\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<mn>8</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 exactly one solution. What is the value of <math alttext=\"c\"><mi>c</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math>. A quadratic equation in 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 exactly one solution when its 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 equal to <math alttext=\"0\"><mn>0</mn>\n</math>. In the given equation, <math alttext=\"minus 16 x squared minus 8 x plus c equals 0\"><mo>-</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>, <math alttext=\"a equals negative 16\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>16</mn>\n</mrow>\n</math> and <math alttext=\"b equals negative 8\"><mrow>\n\t<mi>b</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>8</mn>\n</mrow>\n</math>. Substituting <math alttext=\"negative 16\"><mo>-</mo><mn>16</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"negative 8\"><mo>-</mo><mn>8</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in&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> yields&nbsp;<math alttext=\"left parenthesis negative 8 right parenthesis squared minus 4 left parenthesis negative 16 right parenthesis left parenthesis c right parenthesis\"><msup><mfenced><mrow><mo>-</mo><mn>8</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>16</mn></mrow></mfenced><mfenced><mi>c</mi></mfenced></math>, or <math alttext=\"64 plus 64 c\"><mn>64</mn><mo>+</mo><mn>64</mn><mi>c</mi></math>. Since the given equation has exactly one solution, <math alttext=\"64 plus 64 c equals 0\"><mn>64</mn><mo>+</mo><mn>64</mn><mi>c</mi><mo>=</mo><mn>0</mn></math>. Subtracting <math alttext=\"64\"><mn>64</mn>\n</math> from both sides of this equation yields <math alttext=\"64 c equals negative 64\"><mn>64</mn><mi>c</mi><mo>=</mo><mo>-</mo><mn>64</mn></math>. Dividing both sides of this equation by <math alttext=\"64\"><mn>64</mn>\n</math> yields <math alttext=\"c equals negative 1\"><mrow>\n\t<mi>c</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>1</mn>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"c\"><mi>c</mi>\n</math> is <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math>.</p>","correct_answer":["-1"],"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}}