{"id":"cb_question_bank:7237357e-1316-4960-83c9-b57f1ca58376","source_id":"cb_question_bank","source_item_id":"7237357e-1316-4960-83c9-b57f1ca58376","external_id":"52397fe6-3bd5-4193-935e-114ea31cb4b0","ibn":null,"question_id":"104bff62","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":"StartFraction x squared Over StartRoot x squared minus c squared EndRoot EndFraction equals StartFraction c squared Over StartRoot x squared minus c squared EndRoot EndFraction plus 39 In the given equation, c is a positive constant. Which of the following is one of the solutions to the given equation?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"StartFraction x squared Over StartRoot x squared minus c squared EndRoot EndFraction equals StartFraction c squared Over StartRoot x squared minus c squared EndRoot EndFraction plus 39\"><mrow>\n\t<mrow>\n\t\t<mfrac>\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\t<msqrt>\n\t\t\t\t<mrow>\n\t\t\t\t\t<msup>\n\t\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t\t<mn>2</mn>\n\t\t\t\t\t</msup>\n\t\t\t\t\t<mo>-</mo>\n\t\t\t\t\t<msup>\n\t\t\t\t\t\t<mi>c</mi>\n\t\t\t\t\t\t<mn>2</mn>\n\t\t\t\t\t</msup>\n\t\t\t\t</mrow>\n\t\t\t</msqrt>\n\t\t</mfrac>\n\t</mrow>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mrow>\n\t\t\t<mfrac>\n\t\t\t\t<msup>\n\t\t\t\t\t<mi>c</mi>\n\t\t\t\t\t<mn>2</mn>\n\t\t\t\t</msup>\n\t\t\t\t<msqrt>\n\t\t\t\t\t<mrow>\n\t\t\t\t\t\t<msup>\n\t\t\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t\t\t<mn>2</mn>\n\t\t\t\t\t\t</msup>\n\t\t\t\t\t\t<mo>-</mo>\n\t\t\t\t\t\t<msup>\n\t\t\t\t\t\t\t<mi>c</mi>\n\t\t\t\t\t\t\t<mn>2</mn>\n\t\t\t\t\t\t</msup>\n\t\t\t\t\t</mrow>\n\t\t\t\t</msqrt>\n\t\t\t</mfrac>\n\t\t</mrow>\n\t\t<mo>+</mo>\n\t\t<mn>39</mn>\n\t</mrow>\n</mrow>\n</math></p>\n<p style=\"text-align: left;\">In the given equation, <math alttext=\"c\"><mi>c</mi>\n</math> is a positive constant. Which of the following is one of the solutions to the given equation?</p>","rationale_html":"<p>Choice D is correct. If <math alttext=\"x squared minus c squared less than or equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup><mo>≤</mo><mn>0</mn></math>, then neither side of the given equation is defined and there can be no solution. Therefore, <math alttext=\"x squared minus c squared greater than 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></math>. Subtracting&nbsp;<math alttext=\"StartFraction c squared Over StartRoot x squared minus c squared EndRoot EndFraction\"><mfrac><msup><mi>c</mi><mn>2</mn></msup><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></mfrac></math> from both sides of the given equation yields <math alttext=\"StartFraction x squared Over StartRoot x squared minus c squared EndRoot EndFraction minus StartFraction c squared Over StartRoot x squared minus c squared EndRoot EndFraction equals 39\"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></mfrac><mo>-</mo><mfrac><msup><mi>c</mi><mn>2</mn></msup><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mn>39</mn></math>, or <math alttext=\"StartFraction x squared minus c squared Over StartRoot x squared minus c squared EndRoot EndFraction equals 39\"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mn>39</mn></math>. Squaring both sides of this equation yields <math alttext=\"left parenthesis StartFraction x squared minus c squared Over StartRoot x squared minus c squared EndRoot EndFraction right parenthesis squared equals 39 squared\"><msup><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math>, or <math alttext=\"StartFraction left parenthesis x squared minus c squared right parenthesis left parenthesis x squared minus c squared right parenthesis Over x squared minus c squared EndFraction equals 39 squared\"><mfrac><mrow><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math>. Since <math alttext=\"x squared minus c squared\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></math> is positive and, therefore, nonzero, the expression&nbsp;<math alttext=\"StartFraction x squared minus c squared Over x squared minus c squared EndFraction\"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></math> is defined and equivalent to <math alttext=\"1\"><mn>1</mn>\n</math>. It follows that the equation&nbsp;<math alttext=\"StartFraction left parenthesis x squared minus c squared right parenthesis left parenthesis x squared minus c squared right parenthesis Over x squared minus c squared EndFraction equals 39 squared\"><mfrac><mrow><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math> can be rewritten as&nbsp;<math alttext=\"left parenthesis StartFraction x squared minus c squared Over x squared minus c squared EndFraction right parenthesis left parenthesis x squared minus c squared right parenthesis equals 39 squared\"><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"left parenthesis 1 right parenthesis left parenthesis x squared minus c squared right parenthesis equals 39 squared\"><mfenced><mn>1</mn></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math>, which is equivalent to <math alttext=\"x squared minus c squared equals 39 squared\"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><msup><mn>39</mn><mn>2</mn></msup></math>. Adding&nbsp;<math alttext=\"c squared\"><msup><mi>c</mi><mn>2</mn></msup></math> to both sides of this equation yields <math alttext=\"x squared equals c squared plus 39 squared\"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mn>39</mn><mn>2</mn></msup></math>. Taking the square root of both sides of this equation yields two solutions: <math alttext=\"x equals StartRoot c squared plus 39 squared EndRoot\"><mi>x</mi><mo>=</mo><msqrt><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mn>39</mn><mn>2</mn></msup></msqrt></math> and <math alttext=\"x equals minus StartRoot c squared plus 39 squared EndRoot\"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mn>39</mn><mn>2</mn></msup></msqrt></math>. Therefore, of the given choices, <math alttext=\"minus StartRoot c squared plus 39 squared EndRoot\"><mo>-</mo><msqrt><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mn>39</mn><mn>2</mn></msup></msqrt></math> is one of the solutions to the given equation.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</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>","correct_answer":["D"],"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":[{"label":"A","content_html":"<p style=\"text-align: left;\"><math alttext=\"negative c\"><mrow>\n\t<mo>-</mo>\n\t<mi>c</mi>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"minus c squared minus 39 squared\"><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup><mo>-</mo><msup><mn>39</mn><mn>2</mn></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"minus StartRoot 39 squared minus c squared EndRoot\"><mo>-</mo><msqrt><msup><mn>39</mn><mn>2</mn></msup><mo>-</mo><msup><mi>c</mi><mn>2</mn></msup></msqrt></math></p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\"><math alttext=\"minus StartRoot c squared plus 39 squared EndRoot\"><mo>-</mo><msqrt><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mn>39</mn><mn>2</mn></msup></msqrt></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}}