{"id":"cb_question_bank:2ebf92ab-c1ac-43ad-ba68-f1bb5050e9f7","source_id":"cb_question_bank","source_item_id":"2ebf92ab-c1ac-43ad-ba68-f1bb5050e9f7","external_id":"baae72cd-ec86-42e7-81a6-5f31c096baea","ibn":null,"question_id":"ea6d05bb","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":"The expression left parenthesis 3 x minus 23 right parenthesis left parenthesis 19 x plus 6 right parenthesis is equivalent to the expression a x squared plus b x plus c , where a , b , and c are constants. What is the value of b ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The expression&nbsp;<math alttext=\"left parenthesis 3 x minus 23 right parenthesis left parenthesis 19 x plus 6 right parenthesis\"><mfenced><mrow><mrow><mn>3</mn></mrow><mi>x</mi><mo>-</mo><mrow><mn>23</mn></mrow></mrow></mfenced><mfenced><mrow><mrow><mn>19</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>6</mn></mrow></mrow></mfenced></math> is equivalent to the expression&nbsp;<math alttext=\"a x squared plus b x plus c\"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants. What is the value of <math alttext=\"b\"><mi>b</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 419\"><mo>-</mo><mn>419</mn>\n</math>. It's given that the expression&nbsp;<math alttext=\"left parenthesis 3 x minus 23 right parenthesis left parenthesis 19 x plus 6 right parenthesis\"><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>23</mn></mrow></mfenced><mfenced><mrow><mn>19</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced></math> is equivalent to the expression <math alttext=\"a x squared plus b x plus c\"><mrow>\n\t<mrow>\n\t\t<mi>a</mi>\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<mi>b</mi>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>+</mo>\n\t<mi>c</mi>\n</mrow>\n</math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants. Applying the distributive property to the given expression, <math alttext=\"left parenthesis 3 x minus 23 right parenthesis left parenthesis 19 x plus 6 right parenthesis\"><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>23</mn></mrow></mfenced><mfenced><mrow><mn>19</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfenced></math>, yields&nbsp;<math alttext=\"left parenthesis 3 x right parenthesis left parenthesis 19 x right parenthesis plus left parenthesis 3 x right parenthesis left parenthesis 6 right parenthesis minus left parenthesis 23 right parenthesis left parenthesis 19 x right parenthesis minus left parenthesis 23 right parenthesis left parenthesis 6 right parenthesis\"><mfenced><mrow><mn>3</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>19</mn><mi>x</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><mi>x</mi></mrow></mfenced><mfenced><mn>6</mn></mfenced><mo>-</mo><mfenced><mn>23</mn></mfenced><mfenced><mrow><mn>19</mn><mi>x</mi></mrow></mfenced><mo>-</mo><mfenced><mn>23</mn></mfenced><mfenced><mn>6</mn></mfenced></math>, which can be rewritten as&nbsp;<math alttext=\"57 x squared plus 18 x minus 437 x minus 138\"><mn>57</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>18</mn><mi>x</mi><mo>-</mo><mn>437</mn><mi>x</mi><mo>-</mo><mn>138</mn></math>. Combining like terms yields&nbsp;<math alttext=\"57 x squared minus 419 x minus 138\"><mn>57</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>419</mn><mi>x</mi><mo>-</mo><mn>138</mn></math>. Since this expression is equivalent to <math alttext=\"a x squared plus b x plus c\"><mrow>\n\t<mrow>\n\t\t<mi>a</mi>\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<mi>b</mi>\n\t\t<mi>x</mi>\n\t</mrow>\n\t<mo>+</mo>\n\t<mi>c</mi>\n</mrow>\n</math>, it follows that the value of <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"negative 419\"><mo>-</mo><mn>419</mn>\n</math>.</p>","correct_answer":["-419"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.822000+00:00","source_updated_at":"2023-08-02T20:25:59.822000+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. 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