{"id":"cb_question_bank:6534ed52-4b2a-4171-8e4e-1ad5482678d4","source_id":"cb_question_bank","source_item_id":"6534ed52-4b2a-4171-8e4e-1ad5482678d4","external_id":null,"ibn":"00489-DC","question_id":"a0b4103e","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The expression one-third x squared, minus 2 can be rewritten as one-third times, open parenthesis, x minus k, close parenthesis, times, open parenthesis, x plus k, close parenthesis , where k is a positive constant. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">The expression <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3f348a2f3a5e5606de18208c7b71f3eff8d635721dd1836d8b026032f08265dc\" alt=\"one-third x squared, minus 2 \"></span></span> can be rewritten as <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a6d624292c4417aaa0c48e84389651535905cd0ce6231960beb8ab257e528f67\" alt=\"one-third times, open parenthesis, x minus k, close parenthesis, times, open parenthesis, x plus k, close parenthesis\"></span></span>, where <span class=\"italic\">k</span> is a positive constant. What is the value of <span class=\"italic\">k</span> ?</p>\n","rationale_html":"<p>Choice D is correct. Factoring out the coefficient <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fce26b7f770543c2e33190a4ea311aa1f4d8da39df9b43ef0ea5dfff882a8571\" alt=\"one third\"></span>, the given expression can be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/037c7a86b0da5e7b9e057db847ddf557e2111ffd48f6dd5792399ab6abc0d8cb\" alt=\"one third times, open parenthesis, x squared, minus 6, close parenthesis\"></span>. The expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f8b7f750b534ffbbe4b6aec2dffb03eba95da47aa5eda1096f9b826592b8819\" alt=\"x squared, minus 6\"></span> can be approached as a difference of squares and rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9de173ccff71bb080cd033b2939e8b0e469f761e50985cb82a0581adabd6cdcb\" alt=\"open parenthesis, x minus the square root of 6, close parenthesis, times, open parenthesis, x plus the square root of 6, close parenthesis\"></span>. Therefore, <span class=\"italic\">k</span> must be <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d3019e6233dfe8b588485e9bb1b7570c20bad9a69673f345ca59dbb3b63d83a7\" alt=\"the square root of 6\"></span>.</p><p>Choice A is incorrect. If <span class=\"italic\">k</span> were 2, then the expression given would be rewritten as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/164442f3dba0517b154fa728a20749bbcda13a4d77e19042e2ae96558b3106a8\" alt=\"one third times, open parenthesis, x minus 2, close parenthesis, times, open parenthesis, x plus 2, close parenthesis\"></span>, which is equivalent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9fa80ecdb5021b574a318e604473daba232760f53a11f332b6f6fe3aadf80bfe\" alt=\"one third, x squared, minus four thirds\"></span>, not <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3f348a2f3a5e5606de18208c7b71f3eff8d635721dd1836d8b026032f08265dc\" alt=\"one third, x squared, minus 2\"></span>.</p><p>Choice B is incorrect. This may result from incorrectly factoring the expression and finding <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d7a0be4c9bae848cfb501b384e66fd7278ed2a809ae0f219d495706b2e8eea70\" alt=\"open parenthesis, x minus 6, close parenthesis, times, open parenthesis, x plus 6, close parenthesis\"></span> as the factored form of the expression. Choice C is incorrect. This may result from incorrectly distributing the <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fce26b7f770543c2e33190a4ea311aa1f4d8da39df9b43ef0ea5dfff882a8571\" alt=\"one third\"></span> and rewriting the expression as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f853e94832bc536e33ccec82b98dcf618910dd50380b6cadcc4623c8e256dc9d\" alt=\"one third times, open parenthesis, x squared, minus 2, close parenthesis\"></span>.</p><p></p>\n","correct_answer":["D"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.614000+00:00","source_updated_at":"2023-08-02T20:25:59.614000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:right \">2</span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:right \">6</span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/08a502775ee61c94d766bfb27ce5bf7f1880bb219049fa6c9a288b0a1d2dca8a\" alt=\"the square root of 2\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d3019e6233dfe8b588485e9bb1b7570c20bad9a69673f345ca59dbb3b63d83a7\" alt=\"the square root of 6\"></span></span>\n          </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}}