{"id":"cb_question_bank:c5d6c8d1-532a-4f76-bea0-e831a9915d95","source_id":"cb_question_bank","source_item_id":"c5d6c8d1-532a-4f76-bea0-e831a9915d95","external_id":null,"ibn":"08128-DC","question_id":"f89e1d6f","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"If a equals, c plus d , which of the following is equivalent to the expression x squared, minus c squared, minus 2 c d, minus d squared ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">If <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0e71377148eb1ae686968c9f1b2f6ae7477fdc603a2805bbb645450512e7a793\" alt=\"a equals, c plus d\"></span></span>, which of the following is equivalent to the expression <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/819bd7f5f153f5e89127e59549da0e6cf807ee8ac0746aac39c700931d6ee118\" alt=\"x squared, minus c squared, minus 2 c d, minus d squared\"></span></span>?</p>\n","rationale_html":"<p>Choice C is correct. Factoring –1 from the second, third, and fourth terms gives <span class=\"italic\">x</span><sup>2 </sup>– <span class=\"italic\">c</span><sup>2 </sup>– 2<span class=\"italic\">cd </span>– <span class=\"italic\">d</span><sup>2 </sup>= <span class=\"italic\">x</span><sup>2 </sup>– (<span class=\"italic\">c</span><sup>2 </sup>+ 2<span class=\"italic\">cd </span>+ <span class=\"italic\">d</span><sup>2</sup>). The expression <span class=\"italic\">c</span><sup>2</sup> + 2<span class=\"italic\">cd </span>+ <span class=\"italic\">d</span><sup>2 </sup>is the expanded form of a perfect square: <span class=\"italic\">c</span><sup>2</sup> + 2<span class=\"italic\">cd</span> + <span class=\"italic\">d</span><sup>2</sup> = (<span class=\"italic\">c </span>+ <span class=\"italic\">d</span>)<sup>2</sup>. Therefore, <span class=\"italic\">x</span><sup>2 </sup>– (<span class=\"italic\">c</span><sup>2 </sup>+ 2<span class=\"italic\">cd </span>+ <span class=\"italic\">d</span><sup>2</sup>) = <span class=\"italic\">x</span><sup>2</sup> – (<span class=\"italic\">c </span>+ <span class=\"italic\">d</span>)<sup>2</sup>. Since <span class=\"italic\">a</span> = <span class=\"italic\">c</span> + <span class=\"italic\">d</span>, <span class=\"italic\">x</span><sup>2 </sup>– (<span class=\"italic\">c </span>+ <span class=\"italic\">d</span>)<sup>2 </sup>= <span class=\"italic\">x</span><sup>2 </sup>– <span class=\"italic\">a</span><sup>2</sup>. Finally, because <span class=\"italic\">x</span><sup>2 </sup>– <span class=\"italic\">a</span><sup>2</sup> is the difference of squares, it can be expanded as <span class=\"italic\">x</span><sup>2 </sup>– <span class=\"italic\">a</span><sup>2 </sup>= (<span class=\"italic\">x </span>+ <span class=\"italic\">a</span>)(<span class=\"italic\">x </span>– <span class=\"italic\">a</span>).</p><p>Choices A and B are incorrect and may be the result of making an error in factoring the difference of squares <span class=\"italic\">x</span><sup>2 </sup>– <span class=\"italic\">a</span><sup>2</sup>. Choice D is incorrect and may be the result of incorrectly combining terms.</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.634000+00:00","source_updated_at":"2023-08-02T20:25:59.634000+00:00","options":[{"label":"A","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6703dcf253305f7b921cd1c3da37b0d1368ae9425650847494070b1ab53e478e\" alt=\"open parenthesis, x plus a, close parenthesis, squared\"></span></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b8efc31a3fedd0b9b0f039246ac0fd9c502763f847e22987a0fcc85bf0520201\" alt=\"open parenthesis, x minus a, close parenthesis, squared\"></span></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/011e9c5b0411f22d558fc2df97780155e963f6649ef76d2240ae30f2249fe6a8\" alt=\"open parenthesis, x plus a, close parenthesis, times, open parenthesis, x minus a, close parenthesis\"></span></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/27f247fc4405ea6ae9be3ce038748af957f4506784f3aacd26f3419eb71c22af\" alt=\"x squared, minus a times x, minus a squared\"></span></span>\n","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}}