{"id":"cb_question_bank:c1f6f617-3f33-4f1e-90d4-c0395a2bcc7f","source_id":"cb_question_bank","source_item_id":"c1f6f617-3f33-4f1e-90d4-c0395a2bcc7f","external_id":null,"ibn":"027180-DC","question_id":"16de54c7","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"M","score_band":4,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"If the given expression is rewritten in the form open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis , where k is a constant, what is the value of k ?","stimulus":"2, x squared, plus 5 x, minus 12","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/8822e7a27da6d0fd4d97f768d2c4bacf320ca957baa18300201ca944b852c0e4\" alt=\"2, x squared, plus 5 x, minus 12\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">If the given expression is rewritten in the form <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ea589575adfb70f2d875c6cd4f7eeaf0ad5d09c50413cbea2335a5ddbb7d66a7\" alt=\"open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis\"></span></span>, where <span class=\"italic\">k</span> is a constant, what is the value of <span class=\"italic\">k</span>&nbsp;?</p>\n","rationale_html":"<p>The correct answer is 4. It’s given that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8822e7a27da6d0fd4d97f768d2c4bacf320ca957baa18300201ca944b852c0e4\" alt=\"2 x squared, plus 5 x, minus 12 \"></span> can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ea589575adfb70f2d875c6cd4f7eeaf0ad5d09c50413cbea2335a5ddbb7d66a7\" alt=\"open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis\"></span>; it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/77f4db47b1ab8b41117461e6ddf091d8260a90a1b4180dbd0cf148838074ca11\" alt=\"open parenthesis, 2 x minus 3, close parenthesis, times, open parenthesis, x plus k, close parenthesis, equals, 2 x squared, plus 5 x, minus 12\"></span>. Expanding the left-hand side of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/15087dc1c2985e8736c6e9c1736a8de74b8200ce279b9839274971a68a50bbb4\" alt=\"2 x squared, plus 2 k x, minus 3 x, minus 3 k, equals, 2 x squared, plus 5 x, minus 12\"></span>. Subtracting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9d3002544060950eabfee49e246e233950803b8d430beefea3021ec82dbaef28\" alt=\"2 x squared \"></span> from both sides of this equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9c9c627568b8997c225a318368ead1ab86f4f7eaeb0f55d446593e69affaf9ef\" alt=\"2 k x minus 3 x, minus 3 k, equals, 5 x minus 12\"></span>. Using properties of equality, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/696a1f5e1eef223011721448918635bef0d2faf3ead7eab9f3dd658048f85ffd\" alt=\"2 k x minus 3 x, equals 5 x\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/862fa2ad7cba586332ceb7d67c38a898fac71a25346db7c23f1c4f5043128ec2\" alt=\"negative 3 k, equals negative 12\"></span>. Either equation can be solved for <span class=\"italic\">k</span>. Dividing both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/862fa2ad7cba586332ceb7d67c38a898fac71a25346db7c23f1c4f5043128ec2\" alt=\"negative 3 k, equals negative 12\"></span> by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b62558094601a23d6fe95ffbc037ca0c41dd10264b6e51f4163b9ed575c135a3\" alt=\"negative 3\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/849aab7839d4a1d171c08404fe638c08372999f0ba3b22481d50c825d6363472\" alt=\"k equals 4\"></span>. The equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/696a1f5e1eef223011721448918635bef0d2faf3ead7eab9f3dd658048f85ffd\" alt=\"2 k x minus 3 x, equals 5 x \"></span> can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6dad1e37d1dd2e5e3087d1519eb59ca7fb96700cea82d0fc7674404f00135b2f\" alt=\"x times, open parenthesis, 2 k minus 3, close parenthesis, equals 5 x\"></span>. It follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f580d0e11630dad66c8653126e57c12d621bc79b6451789c3a9d4ea546a8e20c\" alt=\"2 k minus 3, equals 5\"></span>. Solving this equation for <span class=\"italic\">k</span> also yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/849aab7839d4a1d171c08404fe638c08372999f0ba3b22481d50c825d6363472\" alt=\"k equals 4\"></span>. Therefore, the value of <span class=\"italic\">k</span> is 4.</p>\n","correct_answer":["4"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.621000+00:00","source_updated_at":"2023-08-02T20:25:59.621000+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}}