{"id":"cb_question_bank:0e3f3d4a-eebf-4ae0-8012-32c76531dec0","source_id":"cb_question_bank","source_item_id":"0e3f3d4a-eebf-4ae0-8012-32c76531dec0","external_id":"8acbff18-5c28-43c6-85f6-a6d5078b92cd","ibn":null,"question_id":"fc3d783a","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":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the x y -plane, a line with equation 2 y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = - 4 x 2 + b x , where b is a positive constant, what is the value of b ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In the <math><mrow>\n\t<mi>x</mi>\n\t<mi>y</mi>\n</mrow>\n</math>-plane, a line with equation <math><mrow>\n\t<mrow>\n\t\t<mn>2</mn>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>4.5</mn>\n</mrow>\n</math> intersects a parabola at exactly one point. If the parabola has equation <math><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mo>-</mo>\n\t\t\t<mn>4</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</mrow>\n</mrow>\n</math>, where <math><mi>b</mi>\n</math> is a positive constant, what is the value of <math><mi>b</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"6\"><mn>6</mn>\n</math>. It’s given that a line with equation <math alttext=\"2 y equals 4 period 5\"><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn></math>&nbsp;intersects a parabola with equation <math alttext=\"y equals minus 4 x squared plus b x\"><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi></math>, where <math alttext=\"b\"><mi>b</mi>\n</math> is a positive constant, at exactly one point in the <em>xy</em>-plane. It follows that the system of equations consisting of <math alttext=\"2 y equals 4 period 5\"><mn>2</mn><mi>y</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn></math>&nbsp;and&nbsp;<math alttext=\"y equals minus 4 x squared plus b x\"><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi></math> has exactly one solution. Dividing both sides of the equation of the line by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"y equals 2 period 25\"><mi>y</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>25</mn></math>. Substituting <math alttext=\"2.25\"><mn>2.25</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation of the parabola yields <math alttext=\"2 period 25 equals minus 4 x squared plus b x\"><mn>2</mn><mo>.</mo><mn>25</mn><mo>=</mo><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi></math>. Adding <math alttext=\"4 x squared\"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></math> and subtracting <math alttext=\"b x\"><mi>b</mi><mi>x</mi></math> from both sides of this equation yields <math alttext=\"4 x squared minus b x plus 2 period 25 equals 0\"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>b</mi><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>25</mn><mo>=</mo><mn>0</mn></math>. A quadratic equation in the form of <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 exactly one solution when the discriminant, <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 equal to zero. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"2.25\"><mn>2.25</mn>\n</math> for <math alttext=\"c\"><mi>c</mi>\n</math> in the expression <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> and setting this expression equal to <math alttext=\"0\"><mn>0</mn>\n</math> yields <math alttext=\"b squared minus 4 left parenthesis 4 right parenthesis left parenthesis 2 period 25 right parenthesis equals 0\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>(</mo><mn>4</mn><mo>)</mo><mo>(</mo><mn>2</mn><mo>.</mo><mn>25</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>, or <math alttext=\"b squared minus 36 equals 0\"><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>36</mn><mo>=</mo><mn>0</mn></math>. Adding <math alttext=\"36\"><mn>36</mn>\n</math> to each side of this equation yields <math alttext=\"b squared equals 36\"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>36</mn></math>. Taking the square root of each side of this equation yields <math alttext=\"b equals plus or minus 6\"><mi>b</mi><mo>=</mo><mo>±</mo><mn>6</mn></math>. It’s given that <math alttext=\"b\"><mi>b</mi>\n</math> is positive, so the value of <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"6\"><mn>6</mn>\n</math>.</p>","correct_answer":["6"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.823000+00:00","source_updated_at":"2023-08-02T20:25:59.823000+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}}