{"id":"cb_question_bank:66e74ab4-772a-4860-8c42-14ecc92f8855","source_id":"cb_question_bank","source_item_id":"66e74ab4-772a-4860-8c42-14ecc92f8855","external_id":"6c2521be-efeb-4288-b874-77e68fac25dd","ibn":null,"question_id":"2cd6b22d","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":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"5 x squared plus 10 x plus 16 equals 0 How many distinct real solutions does the given equation have?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"5 x squared plus 10 x plus 16 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>5</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>10</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>16</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">How many distinct real solutions does the given equation have?</p>","rationale_html":"<p>Choice D is correct. The number of solutions of 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, can be determined by 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>. If the value of the discriminant is positive, then the quadratic equation has exactly two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is negative, then the quadratic equation has zero real solutions. In the given equation, <math alttext=\"5 x squared plus 10 x plus 16 equals 0\"><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></math>, <math alttext=\"a equals 5\"><mi>a</mi><mo>=</mo><mo>&nbsp;</mo><mn>5</mn></math>, <math alttext=\"b equals 10\"><mi>b</mi><mo>=</mo><mn>10</mn></math>, and <math alttext=\"c equals 16\"><mi>c</mi><mo>=</mo><mn>16</mn></math>. Substituting these values for <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> 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 <math alttext=\"left parenthesis 10 right parenthesis squared minus 4 left parenthesis 5 right parenthesis left parenthesis 16 right parenthesis\"><msup><mfenced><mn>10</mn></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>5</mn></mfenced><mfenced><mn>16</mn></mfenced></math>, or <math alttext=\"negative 220\"><mo>-</mo><mn>220</mn></math>. Since the value of its discriminant is negative, the given equation has zero real solutions. Therefore, the number of distinct real solutions the given equation has is zero.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["D"],"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":[{"label":"A","content_html":"<p>Exactly one</p>","ord":0},{"label":"B","content_html":"<p>Exactly two</p>","ord":1},{"label":"C","content_html":"<p>Infinitely many</p>","ord":2},{"label":"D","content_html":"<p>Zero</p>","ord":3}],"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}}