{"id":134795,"date":"2023-03-21T16:03:49","date_gmt":"2023-03-21T10:33:49","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=134795"},"modified":"2023-03-21T16:03:49","modified_gmt":"2023-03-21T10:33:49","slug":"if-one-root-of-the-equation-4x-2-%e2%88%922x%ce%b2%e2%88%9240-be-the-reciprocal-of-the-other-then-find-%ce%b2","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/if-one-root-of-the-equation-4x-2-%e2%88%922x%ce%b2%e2%88%9240-be-the-reciprocal-of-the-other-then-find-%ce%b2","title":{"rendered":"If one root of the equation 4x 2 \u22122x+(\u03b2\u22124)=0 be the reciprocal of the other, then find \u03b2"},"content":{"rendered":"<h2>Question :If one root of the equation 4x 2 \u22122x+(\u03b2\u22124)=0 be the reciprocal of the other, then find \u03b2<\/h2>\n<h3>The correct answer :Let the roots of the quadratic equation be \u03b1 and 1\/\u03b1.<\/h3>\n<h3>By Vieta&#8217;s formulas, we know that the sum of the roots of a quadratic equation of the form ax^2 + bx + c = 0 is given by -b\/a, and the product of the roots is given by c\/a.<\/h3>\n<h3>So, we have:<\/h3>\n<h3>\u03b1 + (1\/\u03b1) = 2\u03b2\/4 = \u03b2\/2 (1) \u03b1(1\/\u03b1) = 4-\u03b2\/4 = 1-\u03b2\/2 (2)<\/h3>\n<h3>Multiplying equation (1) by \u03b1, we get:<\/h3>\n<h3>\u03b1^2 + 1 = (\u03b2\/2)\u03b1<\/h3>\n<h3>Multiplying equation (2) by 4\/\u03b1, we get:<\/h3>\n<h3>4 + \u03b2 = 8\/\u03b1 &#8211; 2\u03b2\/\u03b1<\/h3>\n<h3>Substituting \u03b1 = 1\/\u03b1 from equation (1), we get:<\/h3>\n<h3>4 + \u03b2 = 8\u03b1 &#8211; 2\u03b2\u03b1 4 + \u03b2 = 8(\u03b1 &#8211; \u03b1^2)<\/h3>\n<h3>Using the identity \u03b1 &#8211; \u03b1^2 = (\u03b1 &#8211; 1\/2)^2 &#8211; 1\/4, we get:<\/h3>\n<h3>4 + \u03b2 = 8[(\u03b1 &#8211; 1\/2)^2 &#8211; 1\/4] 4 + \u03b2 = 8(\u03b1 &#8211; 1\/2)^2 &#8211; 2<\/h3>\n<h3>Substituting \u03b2\/2 for \u03b1 + 1\/\u03b1 from equation (1), we get:<\/h3>\n<h3>4 + \u03b2 = 8(\u03b2\/2 &#8211; 1\/2)^2 &#8211; 2 4 + \u03b2 = 2(\u03b2 &#8211; 1)^2 &#8211; 2 4 + \u03b2 = 2(\u03b2^2 &#8211; 2\u03b2 + 1) &#8211; 2 4 + \u03b2 = 2\u03b2^2 &#8211; 4\u03b2 + 2 2\u03b2^2 &#8211; 5\u03b2 + 2 = 0<\/h3>\n<h3>Using the quadratic formula, we get:<\/h3>\n<h3>\u03b2 = [5 \u00b1 sqrt(25 &#8211; 16)]\/4 \u03b2 = [5 \u00b1 3]\/4<\/h3>\n<h3>Therefore, \u03b2 can be either 2 or 1.<\/h3>\n<h3>So, the possible values of \u03b2 are 1 and 2.<\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Question :If one root of the equation 4x 2 \u22122x+(\u03b2\u22124)=0 be the reciprocal of the other, then find \u03b2 The correct answer :Let the roots of the quadratic equation be \u03b1 and 1\/\u03b1. By Vieta&#8217;s formulas, we know that the sum of the roots of a quadratic equation of the form ax^2 + bx + [&hellip;]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-134795","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134795","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=134795"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134795\/revisions"}],"predecessor-version":[{"id":134796,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134795\/revisions\/134796"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=134795"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=134795"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=134795"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}