{"id":"cb_question_bank:5e6026c1-07e9-460d-b509-3bca75b43429","source_id":"cb_question_bank","source_item_id":"5e6026c1-07e9-460d-b509-3bca75b43429","external_id":"36a520ca-4944-44ee-bfd7-ed88c78436ef","ibn":null,"question_id":"062f86db","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":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"5 x squared minus 37 x minus 24 equals 0 What is the positive solution to the given equation?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"5 x squared minus 37 x minus 24 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>37</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>-</mo>\n\t\t<mn>24</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">What is the positive solution to the given equation?</p>","rationale_html":"<p style=\"text-align: left;\">Choice C is correct. The left-hand side of the given equation can be factored as <math alttext=\"left parenthesis 5 x plus 3 right parenthesis left parenthesis x minus 8 right parenthesis\"><mfenced><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced></math>. Therefore, the given equation, <math alttext=\"5 x squared minus 37 x minus 24 equals 0\"><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>37</mn><mi>x</mi><mo>-</mo><mn>24</mn><mo>=</mo><mn>0</mn></math>, can be written as <math alttext=\"left parenthesis 5 x plus 3 right parenthesis left parenthesis x minus 8 right parenthesis equals 0\"><mfenced><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>. Applying the zero product property to this equation yields&nbsp;<math alttext=\"5 x plus 3 equals 0\"><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> and <math alttext=\"x minus 8 equals 0\"><mi>x</mi><mo>-</mo><mn>8</mn><mo>=</mo><mn>0</mn></math>. Subtracting <math alttext=\"3\"><mn>3</mn>\n</math> from both sides of the equation&nbsp;<math alttext=\"5 x plus 3 equals 0\"><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> yields <math alttext=\"5 x equals negative 3\"><mn>5</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn></math>. Dividing both sides of this equation by <math alttext=\"5\"><mn>5</mn>\n</math> yields <math alttext=\"x equals negative three fifths\"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></math>. Adding <math alttext=\"8\"><mn>8</mn>\n</math> to both sides of the equation&nbsp;<math alttext=\"x minus 8 equals 0\"><mi>x</mi><mo>-</mo><mn>8</mn><mo>=</mo><mn>0</mn></math> yields <math alttext=\"x equals 8\"><mi>x</mi><mo>=</mo><mn>8</mn></math>. Therefore, the two solutions to the given equation, <math alttext=\"5 x squared minus 37 x minus 24 equals 0\"><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>37</mn><mi>x</mi><mo>-</mo><mn>24</mn><mo>=</mo><mn>0</mn></math>, are&nbsp;<math alttext=\"negative three fifths\"><mo>-</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></math> and <math alttext=\"8\"><mn>8</mn>\n</math>. It follows that <math alttext=\"8\"><mn>8</mn>\n</math> is the positive solution to the given equation.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"three fifths\"><mfrac>\n\t<mn>3</mn>\n\t<mn>5</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"3\"><mn>3</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"8\"><mn>8</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"37\"><mn>37</mn>\n</math></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}}