{"id":"cb_question_bank:5240ad4b-bdec-420e-b14c-2fbe39cbe33e","source_id":"cb_question_bank","source_item_id":"5240ad4b-bdec-420e-b14c-2fbe39cbe33e","external_id":null,"ibn":"070620-DC","question_id":"a4f61d75","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":5,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the equation above, a is a constant and a, is greater than 0 . If the equation has two integer solutions, what is a possible value of a ?","stimulus":"x squared, minus a, x, plus 12, equals 0","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/23dc61df1c4dee64d6871a0a177f77c287e251e4bf323821db9e2c6b330dc1ee\" alt=\"x squared, minus a, x, plus 12, equals 0\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the equation above, <span class=\"italic\">a</span> is a constant and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2e1b675bff50b2e5174ef90e2dbd3e6d03eab148944a57c76c89cb483a8fb2aa\" alt=\"a, is greater than 0\"></span></span>. If the equation has two integer solutions, what is a possible value of <span class=\"italic\">a</span> ?</p>\n","rationale_html":"<p>The correct answer is either 7, 8, or 13. Since the given equation has two integer solutions, the expression on the left-hand side of this equation can be factored as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8e2d1b729f15fd9c0952ad75e23ef1196ecabd4bcae92aa8e9e8a5dd75df2be5\" alt=\"open parenthesis, x plus c, close parenthesis, times, open parenthesis, x plus d, close parenthesis\"></span>, where <span class=\"italic\">c</span> and <span class=\"italic\">d</span> are also integers. The product of <span class=\"italic\">c</span> and <span class=\"italic\">d</span> must equal the constant term of the original quadratic expression, which is 12. Additionally, the sum of <span class=\"italic\">c</span> and <span class=\"italic\">d</span> must be a negative number since it’s given that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2e1b675bff50b2e5174ef90e2dbd3e6d03eab148944a57c76c89cb483a8fb2aa\" alt=\"a, is greater than 0\"></span>, but the sign preceding <span class=\"italic\">a</span> in the given equation is negative. The possible pairs of values for <span class=\"italic\">c</span> and <span class=\"italic\">d</span> that satisfy both of these conditions are <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc0609d3d5523eaf8e35097c2a9c872ba31206db61297d0a21597123b1c843d7\" alt=\"negative 4 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b62558094601a23d6fe95ffbc037ca0c41dd10264b6e51f4163b9ed575c135a3\" alt=\"negative 3\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f8ac4e862b233acbee95c501658ff49dd21661d2a931c76a597fe7d20c5d70df\" alt=\"negative 6\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a546b5be2c97283e2d1e93605f422844d5a87f3418a61a7d23019eb596b64f17\" alt=\"negative 2\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/adaa4559a5484d45e4ce9d629baf4e891fe63eb871ec062e9f04e820221ce8d4\" alt=\"negative 12 \"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c1e573ec087416273e06563cee97890abeeed317436562b06a92c8920d4c759d\" alt=\"negative 1\"></span>. Since the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e093f2e6bc37bb9f0879d2fb81ad807441b5ee96e6c834cf6a78899b4dbca09f\" alt=\"negative a \"></span> is the sum of <span class=\"italic\">c</span> and <span class=\"italic\">d</span>, the possible values of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e093f2e6bc37bb9f0879d2fb81ad807441b5ee96e6c834cf6a78899b4dbca09f\" alt=\"negative a \"></span> are <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b4324cd72f5734c92524bc91b488724865c2fcccfa8c39ea5c541ee88e79dfa0\" alt=\"negative 4 plus negative 3, equals negative 7\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/094490574fbc764ed1fb2ea9b66d841c938659fea196bff93ddee7724d55bda5\" alt=\"negative 6 plus negative 2, equals negative 8\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/406daba6bfecaebf248455cf2f203489d40c1245cd09d771ee5ad4689abff8f2\" alt=\"negative 12 plus negative 1, equals negative 13\"></span>. It follows that the possible values of <span class=\"italic\">a</span> are 7, 8, and 13. Note that 7, 8, and 13 are examples of ways to enter a correct answer.</p>\n","correct_answer":["7","8","13"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.631000+00:00","source_updated_at":"2023-08-02T20:25:59.631000+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}}