{"id":154165,"date":"2023-08-03T10:32:52","date_gmt":"2023-08-03T05:02:52","guid":{"rendered":"https:\/\/www.gkseries.com\/blog\/?p=154165"},"modified":"2023-08-03T10:32:54","modified_gmt":"2023-08-03T05:02:54","slug":"suppose-%f0%9d%91%8c-is-distributed-uniformly-in-the-open-interval-16-the-probability-that-the-polynomial-3%f0%9d%91%a52-6%f0%9d%91%a5%f0%9d%91%8c-3%f0%9d%91%8c-6-has-only-real-roots-is","status":"publish","type":"post","link":"https:\/\/www.gkseries.com\/blog\/suppose-%f0%9d%91%8c-is-distributed-uniformly-in-the-open-interval-16-the-probability-that-the-polynomial-3%f0%9d%91%a52-6%f0%9d%91%a5%f0%9d%91%8c-3%f0%9d%91%8c-6-has-only-real-roots-is\/","title":{"rendered":"Suppose \ud835\udc4c is distributed uniformly in the open interval (1,6). The probability that the polynomial 3\ud835\udc652 + 6\ud835\udc65\ud835\udc4c + 3\ud835\udc4c + 6 has only real roots is"},"content":{"rendered":"\n

Q. Suppose \ud835\udc4c is distributed uniformly in the open interval (1,6). The probability that the polynomial 3\ud835\udc652<\/sup> + 6\ud835\udc65\ud835\udc4c + 3\ud835\udc4c + 6 has only real roots is (rounded off to 1 decimal place)<\/p>\n\n\n\n

Solution:<\/p>\n\n\n\n

For a quadratic polynomial ax2 + bx + c = 0. There are three condition:<\/p>\n\n\n\n

b2 – 4ac > 0   {real and distinct root, i.e., two real roots}<\/p>\n\n\n\n

b2 – 4ac = 0     {real and equal roots, i.e., only one real root}<\/p>\n\n\n\n

b2 – 4ac < 0    {imaginary roots}<\/p>\n\n\n\n

Polynomial 3×2 + 6xY + 3Y + 6 has only real roots,<\/p>\n\n\n\n

\u21d2 b2 \u2013 4ax \u2265 0<\/p>\n\n\n\n

\u21d2 (6Y)2 \u2013 4(3) (3Y+ 6) \u2265 0<\/p>\n\n\n\n

\u21d2 Y2 \u2013 Y + 2 \u2265 0<\/p>\n\n\n\n

Y \u2208 (\u2013\u221e, \u2013 1] \u2229 [2, \u221e)<\/p>\n\n\n\n

\u21d2 Y \u2208 [2, 6)<\/p>\n\n\n\n

Since y is uniformly distributed in (1, 6).<\/p>\n\n\n\n

Probability distributed function,<\/p>\n\n\n\n

\u00a0f(Y) = (1\/5), 1 < y > 6<\/p>\n\n\n\n

Hence<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Q. Suppose \ud835\udc4c is distributed uniformly in the open interval (1,6). The probability that the polynomial 3\ud835\udc652 + 6\ud835\udc65\ud835\udc4c + 3\ud835\udc4c + 6 has only real roots is (rounded off to 1 decimal place) Solution: For a quadratic polynomial ax2 + bx + c = 0. There are three condition: b2 – 4ac > 0   […]<\/p>\n","protected":false},"author":419,"featured_media":154168,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[5141],"tags":[5140],"class_list":["post-154165","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-gate","tag-gate-questions"],"_links":{"self":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/users\/419"}],"replies":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/comments?post=154165"}],"version-history":[{"count":1,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154165\/revisions"}],"predecessor-version":[{"id":154169,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/posts\/154165\/revisions\/154169"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media\/154168"}],"wp:attachment":[{"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/media?parent=154165"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/categories?post=154165"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.gkseries.com\/blog\/wp-json\/wp\/v2\/tags?post=154165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}