{"id":164,"date":"2018-09-14T14:12:07","date_gmt":"2018-09-14T19:12:07","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=164"},"modified":"2018-09-21T14:14:51","modified_gmt":"2018-09-21T19:14:51","slug":"problem-2-september-14-2018-probability-both-roots-are-positive-and-real","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2018\/09\/14\/problem-2-september-14-2018-probability-both-roots-are-positive-and-real\/","title":{"rendered":"Problem 2:  September, 14 2018 &#8211; Probability Both Roots are Positive and Real"},"content":{"rendered":"<p><span style=\"font-size: xx-large\"><b>The Probability Both Roots are Positive and Real<\/b><\/span><\/p>\n<p><i>This problem is from Gary Gordon, Lafayette College<\/i><\/p>\n<p>Let <i>f<\/i>(<i>x<\/i>) = <i>x<\/i><sup>2<\/sup> + <i>bx<\/i> + <i>c<\/i> where <i>b<\/i> and <i>c<\/i> are real numbers between -1 and 1. What is the probability that both roots of <i>f<\/i>(<i>x<\/i>) are positive real numbers?<\/p>\n<p>Submit your answers to mathpotw@acu.edu. \u00a0Details for submissions can be found <a href=\"http:\/\/blogs.acu.edu\/mu_sigma\/2018\/08\/28\/problem-of-the-week-competition\/\">here<\/a>.<\/p>\n<p><span style=\"font-size: large\">Solution to <b>The Probability Both Roots are Positive and Real<\/b><\/span><\/p>\n<p>Students submitting correct solutions: \u00a0Wyatt Witemeyer,<\/p>\n<p>Let <i>r<\/i> and <i>s<\/i> be the roots of <i>f<\/i>(<i>x<\/i>), and consider the unit square in the plane with the horizontal axis representing (say) <i>b<\/i> values and the vertical axis representing <i>c<\/i> values.<\/p>\n<p>Since <i>f<\/i>(<i>x<\/i>) = (<i>x &#8211; r<\/i>)(<i>x &#8211; s<\/i>) = <i>x<\/i><sup>2<\/sup> &#8211; (<i>r+s<\/i>)<i>x<\/i> + <i>rs<\/i>, it follows that <i>b<\/i> must be negative and <i>c<\/i> must be positive for both <i>r<\/i> and <i>s<\/i> to be positive. This happens in the top left quadrant of the unit square.<\/p>\n<p>For the roots to both be real, we must have <i>b<\/i><sup>2<\/sup> larger than or equal to 4<i>c<\/i>. The region that satisfies this condition is below the graph of the curve given by <i>c<\/i> = <i>b<\/i><sup>2<\/sup> \/ 4.<\/p>\n<p>By integration of the function <i>x<\/i><sup>2<\/sup> \/ 4, we find the area of the region satisfying both conditions above to be 1\/12. Since the total area of the unit square is 4, the probability that both roots are positive and real is<\/p>\n<p align=\"CENTER\">1\/48.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Probability Both Roots are Positive and Real This problem is from Gary Gordon, Lafayette College Let f(x) = x2 + bx + c where b and c are real numbers between -1 and 1. What is the probability that both roots of f(x) are positive real numbers? Submit your answers to mathpotw@acu.edu. \u00a0Details for [&hellip;]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-164","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/164","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=164"}],"version-history":[{"count":3,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/164\/revisions"}],"predecessor-version":[{"id":168,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/164\/revisions\/168"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}