{"id":"cb_question_bank:2b63154b-a952-4f4b-a353-f5d2c4c4d690","source_id":"cb_question_bank","source_item_id":"2b63154b-a952-4f4b-a353-f5d2c4c4d690","external_id":"6dc78b73-3d3a-47a2-8b1b-c6041a6a7337","ibn":null,"question_id":"f5aa5040","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 xy -plane, a line with equation 2 y equals c for some constant c intersects a parabola at exactly one point. If the parabola has equation y equals minus 2 x squared plus 9 x , what is the value of c ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In the <em>xy</em>-plane, a line with equation <math alttext=\"2 y equals c\"><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<mi>c</mi>\n</mrow>\n</math> for some constant <math alttext=\"c\"><mi>c</mi>\n</math> intersects a parabola at exactly one point. If the parabola has equation <math alttext=\"y equals minus 2 x squared plus 9 x\"><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>2</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<mn>9</mn>\n\t\t\t<mi>x</mi>\n\t\t</mrow>\n\t</mrow>\n</mrow>\n</math>, what is the value of <math alttext=\"c\"><mi>c</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 81 Over 4 EndFraction\"><mfrac><mn>81</mn><mn>4</mn></mfrac></math>. The given linear equation is <math alttext=\"2 y equals c\"><mn>2</mn><mi>y</mi><mo>=</mo><mi>c</mi></math>. Dividing both sides of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields <math alttext=\"y equals StartFraction c Over 2 EndFraction\"><mi>y</mi><mo>=</mo><mfrac><mi>c</mi><mn>2</mn></mfrac></math>. Substituting&nbsp;<math alttext=\"StartFraction c Over 2 EndFraction\"><mfrac><mi>c</mi><mn>2</mn></mfrac></math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation of the parabola yields <math alttext=\"StartFraction c Over 2 EndFraction equals minus 2 x squared plus 9 x\"><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi></math>. Adding&nbsp;<math alttext=\"2 x squared\"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> and <math alttext=\"minus 9 x\"><mrow>\n\t<mo>-</mo>\n\t<mn>9</mn>\n\t<mi>x</mi>\n</mrow>\n</math> to both sides of this equation yields <math alttext=\"2 x squared minus 9 x plus StartFraction c Over 2 EndFraction equals 0\"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math>. Since it’s given that the line and the parabola intersect at exactly one point, the equation <math alttext=\"2 x squared minus 9 x plus StartFraction c Over 2 EndFraction equals 0\"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math> must have exactly one solution. An equation of the form <math alttext=\"upper A x squared plus upper B x plus upper 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=\"upper A\"><mi>A</mi>\n</math>, <math alttext=\"upper B\"><mi>B</mi>\n</math>, and <math alttext=\"upper C\"><mi>C</mi>\n</math> are constants, has exactly one solution when the discriminant, <math alttext=\"upper B squared minus 4 upper A upper C\"><msup><mi>B</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>A</mi><mi>C</mi></math>, is equal to <math alttext=\"0\"><mn>0</mn>\n</math>. In the equation <math alttext=\"2 x squared minus 9 x plus StartFraction c Over 2 EndFraction equals 0\"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mfrac><mi>c</mi><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math>, where <math alttext=\"upper A equals 2\"><mi>A</mi><mo>=</mo><mn>2</mn></math>, <math alttext=\"upper B equals negative 9\"><mi>B</mi><mo>=</mo><mo>-</mo><mn>9</mn></math>, and <math alttext=\"upper C equals StartFraction c Over 2 EndFraction\"><mi>C</mi><mo>=</mo><mfrac><mi>c</mi><mn>2</mn></mfrac></math>, the discriminant is <math alttext=\"left parenthesis negative 9 right parenthesis squared minus 4 left parenthesis 2 right parenthesis left parenthesis StartFraction c Over 2 EndFraction right parenthesis\"><msup><mfenced><mrow><mo>-</mo><mn>9</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mfrac><mi>c</mi><mn>2</mn></mfrac></mfenced></math>. Setting the discriminant equal to <math alttext=\"0\"><mn>0</mn>\n</math> yields <math alttext=\"left parenthesis negative 9 right parenthesis squared minus 4 left parenthesis 2 right parenthesis left parenthesis StartFraction c Over 2 EndFraction right parenthesis equals 0\"><msup><mfenced><mrow><mo>-</mo><mn>9</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mfrac><mi>c</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><mn>0</mn></math>, or <math alttext=\"81 minus 4 c equals 0\"><mn>81</mn><mo>-</mo><mn>4</mn><mi>c</mi><mo>=</mo><mn>0</mn></math>. Adding <math alttext=\"4 c\"><mrow>\n\t<mn>4</mn>\n\t<mi>c</mi>\n</mrow>\n</math> to both sides of this equation yields <math alttext=\"81 equals 4 c\"><mn>81</mn><mo>=</mo><mn>4</mn><mi>c</mi></math>. Dividing both sides of this equation by <math alttext=\"4\"><mn>4</mn>\n</math> yields <math alttext=\"c equals StartFraction 81 Over 4 EndFraction\"><mi>c</mi><mo>=</mo><mfrac><mn>81</mn><mn>4</mn></mfrac></math>. Note that 81/4 and 20.25 are examples of ways to enter a correct answer.</p>","correct_answer":["20.25","81/4"],"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}}