{"id":"cb_question_bank:4e1f9e45-da43-4406-ba07-1fac7c0cce36","source_id":"cb_question_bank","source_item_id":"4e1f9e45-da43-4406-ba07-1fac7c0cce36","external_id":"01383d90-cb64-4777-9f89-45b8dfe8c286","ibn":null,"question_id":"5355c0ef","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.A.","skill":"Equivalent expressions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"0.36 x squared plus 0.63 x plus 1.17 The given expression can be rewritten as a left parenthesis 4 x squared plus 7 x plus 13 right parenthesis , where a is a constant. What is the value of a ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"0.36 x squared plus 0.63 x plus 1.17\"><mrow>\n\t<mrow>\n\t\t<mn>0.36</mn>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>+</mo>\n\t<mrow>\n\t\t<mn>0.63</mn>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>+</mo>\n\t<mn>1.17</mn>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">The given expression can be rewritten as <math alttext=\"a left parenthesis 4 x squared plus 7 x plus 13 right parenthesis\"><mi>a</mi><mfenced><mrow><mrow><mn>4</mn></mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>7</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>13</mn></mrow></mrow></mfenced></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is a constant. What is the value of <math alttext=\"a\"><mi>a</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\".09\"><mo>.09</mo></math>. It's given that the expression&nbsp;<math alttext=\"0.36 x squared plus 0.63 x plus 1.17\"><mn>0.36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>0.63</mn><mi>x</mi><mo>+</mo><mn>1.17</mn></math> can be rewritten as <math alttext=\"a left parenthesis 4 x squared plus 7 x plus 13 right parenthesis\"><mi>a</mi><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>13</mn></mrow></mfenced></math>. Applying the distributive property to the expression&nbsp;<math alttext=\"a left parenthesis 4 x squared plus 7 x plus 13 right parenthesis\"><mi>a</mi><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>13</mn></mrow></mfenced></math> yields <math alttext=\"4 a x squared plus 7 a x plus 13 a\"><mn>4</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>13</mn><mi>a</mi></math>. Therefore, <math alttext=\"0.36 x squared plus 0.63 x plus 1.17\"><mn>0.36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>0.63</mn><mi>x</mi><mo>+</mo><mn>1.17</mn></math> can be rewritten as <math alttext=\"4 a x squared plus 7 a x plus 13 a\"><mn>4</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>13</mn><mi>a</mi></math>. It follows that in the expressions <math alttext=\"0.36 x squared plus 0.63 x plus 1.17\"><mn>0.36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>0.63</mn><mi>x</mi><mo>+</mo><mn>1.17</mn></math> and&nbsp;<math alttext=\"4 a x squared plus 7 a x plus 13 a\"><mn>4</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>13</mn><mi>a</mi></math>, the coefficients of <math alttext=\"x squared\"><msup><mi>x</mi><mn>2</mn></msup></math> are equivalent, the coefficients of <math alttext=\"x\"><mi>x</mi>\n</math> are equivalent, and the constant terms are equivalent. Therefore, <math alttext=\"0.36 equals 4 a\"><mrow>\n\t<mn>0.36</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>4</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math>, <math alttext=\"0.63 equals 7 a\"><mrow>\n\t<mn>0.63</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>7</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math>, and <math alttext=\"1.17 equals 13 a\"><mrow>\n\t<mn>1.17</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>13</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math>. Solving any of these equations for <math alttext=\"a\"><mi>a</mi>\n</math> yields the value of <math alttext=\"a\"><mi>a</mi>\n</math>. Dividing both sides of the equation <math alttext=\"0.36 equals 4 a\"><mrow>\n\t<mn>0.36</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>4</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math> by <math alttext=\"4\"><mn>4</mn>\n</math> yields <math alttext=\"0.09 equals a\"><mrow>\n\t<mn>0.09</mn>\n\t<mo>=</mo>\n\t<mi>a</mi>\n</mrow>\n</math>. Therefore, the value of <math alttext=\"a\"><mi>a</mi>\n</math> is <math alttext=\"0.09\"><mn>0.09</mn>\n</math>. Note that .09 and 9/100 are examples of ways to enter a correct answer.</p>","correct_answer":[".09","9/100"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.821000+00:00","source_updated_at":"2023-08-02T20:25:59.821000+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}}