{"id":"cb_question_bank:46c0885d-19b5-4ccd-a652-11c874714e04","source_id":"cb_question_bank","source_item_id":"46c0885d-19b5-4ccd-a652-11c874714e04","external_id":null,"ibn":"08326-DC","question_id":"7bd10ef3","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":"In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?","stimulus":"2 x squared, minus 4 x, equals t","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math_expression \">\n            <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/962d08e595f685eef31ba895d6033cfe7eff38e23790b12d7a677de7bbb63ebb\" alt=\"2 x squared, minus 4 x, equals t\"></span></span>\n        </p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">In the equation above, <span class=\"italic\">t</span> is a constant. If the equation has no real solutions, which of the following could be the value of <span class=\"italic\">t</span> ?</p>\n","rationale_html":"<p>Choice A is correct. The number of solutions to any quadratic equation in the form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2c869bdc70b1a9ecef53fb3c6051020877ac5c401e6cc5aaf10e36e3b7478c63\" alt=\"a, x squared, plus b x, plus c, equals 0\"></span>, where <span class=\"italic\">a</span>, <span class=\"italic\">b</span>, and <span class=\"italic\">c</span> are constants, can be found by evaluating the expression <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/770fcd78f80fbd0094b76eca7ec89e701f2123034b3eb579acff925815bfecd6\" alt=\"b squared, minus 4 a, c\"></span>, which is called the discriminant. If the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/770fcd78f80fbd0094b76eca7ec89e701f2123034b3eb579acff925815bfecd6\" alt=\"b squared, minus 4 a, c\"></span> is a positive number, then there will be exactly two real solutions to the equation. If the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/770fcd78f80fbd0094b76eca7ec89e701f2123034b3eb579acff925815bfecd6\" alt=\"b squared, minus 4 a, c\"></span> is zero, then there will be exactly one real solution to the equation. Finally, if the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/770fcd78f80fbd0094b76eca7ec89e701f2123034b3eb579acff925815bfecd6\" alt=\"b squared, minus 4 a, c\"></span> is negative, then there will be no real solutions to the equation.</p><p>The given equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/962d08e595f685eef31ba895d6033cfe7eff38e23790b12d7a677de7bbb63ebb\" alt=\"2 x squared, minus 4 x, equals t\"></span> is a quadratic equation in one variable, where <span class=\"italic\">t</span> is a constant. Subtracting <span class=\"italic\">t</span> from both sides of the equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e37ae21890e4a8addddcb6fc8ac445e06bf5c0e3efe7b348c1ae876693b5c53e\" alt=\"2 x squared, minus 4 x, minus t, equals 0\"></span>. In this form, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b8939f0089034fb0b27a6a9b987733739fbdb2c6f74447b4a31584d266b4c797\" alt=\"a, equals 2\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/50b0b5d64f33d75756fbd59a05d0c0348307ba7faa30df6bc52cabc3e2f88a2d\" alt=\"b equals negative 4\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/54118f23574c2f3793633440dc7d0101052806349c2aec49c1ae87486218eed1\" alt=\"c equals negative t\"></span>. The values of <span class=\"italic\">t</span> for which the equation has no real solutions are the same values of <span class=\"italic\">t</span> for which the discriminant of this equation is a negative value. The discriminant is equal to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f740a7c4c6cc3139ad740541c2c2ffd0b6bd2bbeaa528fb6d40d9dd9cb396c11\" alt=\"open parenthesis, negative 4, close parenthesis, squared, minus 4 times 2, times negative t\"></span>; therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1007ba5afdfb5a49fb67c9697a92f97befebfc4592427953035fdab06c2aeae6\" alt=\"open parenthesis, negative 4, close parenthesis, squared, minus, 4 times 2, times negative t, is less than 0\"></span>. Simplifying the left side of the inequality gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ee8da77958127e8407341affda4cf09d2ae0bfd155a20662e76bff181340b250\" alt=\"16 plus 8, t, is less than 0\"></span>. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6ca1e851b844eb5f67e4a469c7a4fa9e93fd3ffbc1cb74f28a2376bdf3de2144\" alt=\"t is less than negative 2\"></span>. Of the values given in the options, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b62558094601a23d6fe95ffbc037ca0c41dd10264b6e51f4163b9ed575c135a3\" alt=\"negative 3\"></span> is the only value that is less than <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a546b5be2c97283e2d1e93605f422844d5a87f3418a61a7d23019eb596b64f17\" alt=\"negative 2\"></span>. Therefore, choice A must be the correct answer.</p><p>Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.</p><p>\n        \n      </p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.635000+00:00","source_updated_at":"2023-08-02T20:25:59.635000+00:00","options":[{"label":"A","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/fcb3b618cff214ac5a09c033eaa100f5ee4fc55b088acb71912333adf75e3f91\" alt=\"negative 3\"></span></span>\n          </span>\n","ord":0},{"label":"B","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b3ef0bab498e581649a76baff7488f24fc7f565702b5178801873c0eccc8d6a5\" alt=\"negative 1\"></span></span>\n          </span>\n","ord":1},{"label":"C","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5ad32153d05f6633d54616a039a723d4ba338db72acbbde018e6a4ef708f7feb\" alt=\"1\"></span></span>\n          </span>\n","ord":2},{"label":"D","content_html":"<span class=\"choice_cell align:right \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/29a80bc46d29335c2c283c2829591bbe1c693bfb845705faf2c15f5ddc25b28f\" alt=\"3\"></span></span>\n          </span>\n","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}}