{"id":"cb_practice_pdf:t10-math-m2-q24","source_id":"cb_practice_pdf","source_item_id":"t10-math-m2-q24","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"mcq","answer_source":"rationale","layout_warning":"Part of this question was typeset outside the text flow — a fraction, subscript or similar — and PDF extraction re-inserted it in the wrong place. The wording is reliable; the symbols may be out of order.","form_id":"cb_practice_pdf:test-10","module_number":2,"position":24,"stem":"−9x^2 +30x+c = 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c? 3","stimulus":"","stimulus_html":null,"stem_html":"<p>−9x^2 +30x+c = 0 In the given equation, c is a constant. The equation has exactly one solution. What is the value of c? 3</p>","rationale_html":"<p>-9 2+30 + =0 Choice C is correct. It’s given that the equation x x c has exactly 2+ + =0 one solution. A quadratic equation of the form ax bx c has exactly one -4 + 2 solution if and only if its discriminant, ac b , is equal to zero. It follows that for =- =30 - 30 the given equation, a 9 and b . Substituting 9 for a and for b into 2-4 302-4 -9 900+36 b ac yields ^ h^ch, or c. Since the discriminant must equal 900+36 =0 36 zero, c . Subtracting c from both sides of this equation yields 900=-36 -36 -25= c. Dividing each side of this equation by yields c. -25 Therefore, the value of c is . Choice A is incorrect. If the value of c is 3, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution. Choice B is incorrect. If the value of c is 0, this would yield a discriminant that is greater than zero. Therefore, the given equation would have two solutions, rather than exactly one solution. Choice D is incorrect. If the -53 value of c is , this would yield a discriminant that is less than zero. Therefore, the given equation would have no real solutions, rather than exactly one solution.</p>","correct_answer":["C"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"<p>0</p>","ord":0},{"label":"B","content_html":"","ord":1},{"label":"C","content_html":"<p>−25 −53</p>","ord":2},{"label":"D","content_html":"","ord":3}],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}