{"id":"cb_question_bank:361aa4cb-9749-46f4-8f36-f330465c74b8","source_id":"cb_question_bank","source_item_id":"361aa4cb-9749-46f4-8f36-f330465c74b8","external_id":"ecc55958-e485-4d44-84c5-3a70884d1fb5","ibn":null,"question_id":"d4950429","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A rectangle has a length of x units and a width of left parenthesis x minus 15 right parenthesis units. If the rectangle has an area of 76 square units, what is the value of x ?","stimulus":"","stimulus_html":null,"stem_html":"<p>A rectangle has a length of <math alttext=\"x\"><mi>x</mi>\n</math> units and a width of <math alttext=\"left parenthesis x minus 15 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>15</mn></mrow></mrow></mfenced></math> units. If the rectangle has an area of <math alttext=\"76\"><mn>76</mn>\n</math> square units, what is the value of <math alttext=\"x\"><mi>x</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, <math alttext=\"x\"><mi>x</mi>\n</math> units, by the given width, <math alttext=\"left parenthesis x minus 15 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced></math> units, yields&nbsp;<math alttext=\"x left parenthesis x minus 15 right parenthesis\"><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced></math> square units. If the rectangle has an area of <math alttext=\"76\"><mn>76</mn>\n</math> square units, it follows that&nbsp;<math alttext=\"x left parenthesis x minus 15 right parenthesis equals 76\"><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mo>=</mo><mn>76</mn></math>, or&nbsp;<math alttext=\"x squared minus 15 x equals 76\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>15</mn><mi>x</mi><mo>=</mo><mn>76</mn></math>. Subtracting <math alttext=\"76\"><mn>76</mn>\n</math> from both sides of this equation yields&nbsp;<math alttext=\"x squared minus 15 x minus 76 equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>15</mn><mi>x</mi><mo>-</mo><mn>76</mn><mo>=</mo><mn>0</mn></math>. Factoring the left-hand side of this equation yields&nbsp;<math alttext=\"left parenthesis x minus 19 right parenthesis left parenthesis x plus 4 right parenthesis equals 0\"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>19</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>. Applying the zero product property to this equation yields two solutions: <math alttext=\"x equals 19\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>19</mn>\n</mrow>\n</math> and <math alttext=\"x equals negative 4\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>4</mn>\n</mrow>\n</math>. Since <math alttext=\"x\"><mi>x</mi>\n</math> is the rectangle’s length, in units, which must be positive, the value of <math alttext=\"x\"><mi>x</mi>\n</math> is <math alttext=\"19\"><mn>19</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This is the width, in units, of the rectangle, not the value of <math alttext=\"x\"><mi>x</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of <math alttext=\"x\"><mi>x</mi>\n</math>.</p>","correct_answer":["B"],"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=\"4\"><mn>4</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"19\"><mn>19</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"23\"><mn>23</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"76\"><mn>76</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}}