{"id":"cb_question_bank:d11caa6c-8bec-4d21-8ffc-9dbb4006c59d","source_id":"cb_question_bank","source_item_id":"d11caa6c-8bec-4d21-8ffc-9dbb4006c59d","external_id":"47af8694-05ad-4849-a1fb-69bbee9cc524","ibn":null,"question_id":"6011a3f8","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.B.","skill":"Nonlinear equations in one variable and systems of equations in two variables","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"64 x squared plus b x plus 25 equals 0 In the given equation, b is a constant. For which of the following values of b will the equation have more than one real solution?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"64 x squared plus b x plus 25 equals 0\"><mrow>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mn>64</mn>\n\t\t\t<msup>\n\t\t\t\t<mi>x</mi>\n\t\t\t\t<mn>2</mn>\n\t\t\t</msup>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mrow>\n\t\t\t<mi>b</mi>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>25</mn>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"b\"><mi>b</mi>\n</math> is a constant. For which of the following values of <math alttext=\"b\"><mi>b</mi>\n</math> will the equation have more than one real solution?</p>","rationale_html":"<p>Choice A is correct. A quadratic equation of the form <math alttext=\"a x squared plus b x plus c equals 0\"><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></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, has either no real solutions, exactly one real solution, or exactly two real solutions. That is, for the given equation to have more than one real solution, it must have exactly two real solutions. When the value of the discriminant, or <math alttext=\"b squared minus 4 a c\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mi>c</mi></math>, is greater than 0, the given equation has exactly two real solutions. In the given equation, <math alttext=\"64 x squared plus b x plus 25 equals 0\"><mn>64</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mn>25</mn><mo>=</mo><mn>0</mn></math>,&nbsp;<math alttext=\"a equals 64\"><mi>a</mi><mo>=</mo><mn>64</mn></math> and <math alttext=\"c equals 25\"><mi>c</mi><mo>=</mo><mn>25</mn></math>. Therefore, the given equation has exactly two real solutions when <math alttext=\"left parenthesis b right parenthesis squared minus 4 left parenthesis 64 right parenthesis left parenthesis 25 right parenthesis greater than 0\"><msup><mfenced><mi>b</mi></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>64</mn></mfenced><mfenced><mn>25</mn></mfenced><mo>&gt;</mo><mn>0</mn></math>, or <math alttext=\"b squared minus 6,400 greater than 0\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>6,400</mn><mo>&gt;</mo><mn>0</mn></math>. Adding <math alttext=\"6,400\"><mn>6,400</mn>\n</math> to both sides of this inequality yields <math alttext=\"b squared greater than 6,400\"><msup><mi>b</mi><mn>2</mn></msup><mo>&gt;</mo><mn>6,400</mn></math>. Taking the square root of both sides of <math alttext=\"b squared greater than 6,400\"><msup><mi>b</mi><mn>2</mn></msup><mo>&gt;</mo><mn>6,400</mn></math> yields two possible inequalities: <math alttext=\"b less than negative 80\"><mi>b</mi><mo>&lt;</mo><mo>-</mo><mn>80</mn></math> or <math alttext=\"b greater than 80\"><mi>b</mi><mo>&gt;</mo><mn>80</mn></math>. Of the choices, only choice A satisfies <math alttext=\"b less than negative 80\"><mi>b</mi><mo>&lt;</mo><mo>-</mo><mn>80</mn></math> or <math alttext=\"b greater than 80\"><mi>b</mi><mo>&gt;</mo><mn>80</mn></math>.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.824000+00:00","source_updated_at":"2023-08-02T20:25:59.824000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"negative 91\"><mo>-</mo><mn>91</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"negative 80\"><mo>-</mo><mn>80</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"5\"><mn>5</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"40\"><mn>40</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}}