{"id":6681,"date":"2021-10-19T19:41:51","date_gmt":"2021-10-19T14:11:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6681"},"modified":"2021-11-27T23:25:22","modified_gmt":"2021-11-27T17:55:22","slug":"nature-of-roots-of-quadratic-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/","title":{"rendered":"Nature of Roots of Quadratic Equation"},"content":{"rendered":"

Here you will learn how to find the nature of roots of quadratic equation using discriminant with examples.<\/p>\n

Let’s begin –<\/p>\n

Nature of Roots of Quadratic Equation<\/h2>\n

(a) Consider the quadratic equation \\(ax^2 + bx + c\\) = 0 where a, b, c \\(\\in\\) R & a \\(\\ne\\) 0 , then<\/p>\n

x = \\(-b \\pm \\sqrt{D}\\over 2a\\)    where D = \\(b^2 – 4ac\\)<\/p>\n

So, a quadratic equation \\(ax^2 + bx + c\\) = 0 <\/p>\n

\n

(i) has no real roots if D < 0.<\/p>\n

(ii) has two equal real roots if D = 0.<\/p>\n

(iii) has two distinct real roots if D > 0.<\/p>\n

(iv) has real roots if D \\(\\ge\\) 0.<\/p>\n

(v) If p + iq is one of the root of a quadratic equation, then the other root must be the conjugate p – iq & vice versa. (p, q \\(\\in\\) R & i = \\(\\sqrt{-1}\\)).<\/p>\n<\/blockquote>\n

(b) Consider the quadratic equation \\(ax^2 + bx + c\\) = 0 where a, b, c \\(\\in\\) Q & a \\(\\in\\) 0, then :<\/p>\n

\n

(i) If D is a perfect square, then roots are rational.<\/p>\n

(ii) if \\(\\alpha\\) = p + \\(\\sqrt{q}\\) is one root in this case, (where p is rational & \\(\\sqrt{q}\\) is a surd) then other root will be p – \\(\\sqrt{q}\\).<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Examine, whether the following equations have real roots :<\/p>\n

(i) \\(x^2 + x + 1\\) = 0<\/p>\n

(ii) \\(3x^2 + 2x – 1\\) = 0<\/p>\n

Solution<\/span><\/strong> : <\/p>\n

(i) We have, \\(x^2 + x + 1\\) = 0<\/p>\n

Here,  a = 1,  b = 1,  c = 1<\/p>\n

Therefore, Discriminant D = \\(b^2 – 4ac\\) = \\((1)^2 – 4(1)(1)\\) = -3 < 0.<\/p>\n

Since, D < 0, the equation has no real roots.<\/p>\n

(ii) We have, \\(3x^2 + 2x – 1\\) = 0<\/p>\n

Here,  a = 3,  b = 2,  c = -1<\/p>\n

Therefore, Discriminant D = \\(b^2 – 4ac\\) = \\((2)^2 – 4(3)(-1)\\) = 16 > 0.<\/p>\n

Since, D > 0, the equation has real roots.<\/p>\n\n\n

\n
Next – Relation Between Roots and Coefficients of Quadratic Equation<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Solve Quadratic Equation Using Quadratic Formula<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the nature of roots of quadratic equation using discriminant with examples. Let’s begin – Nature of Roots of Quadratic Equation (a) Consider the quadratic equation \\(ax^2 + bx + c\\) = 0 where a, b, c \\(\\in\\) R & a \\(\\ne\\) 0 , then x = \\(-b \\pm …<\/p>\n

Nature of Roots of Quadratic Equation<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[42],"tags":[562,561,560],"yoast_head":"\nNature of Roots of Quadratic Equation - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Nature of Roots of Quadratic Equation - Mathemerize\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-19T14:11:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2021-11-27T17:55:22+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Nature of Roots of Quadratic Equation\",\"datePublished\":\"2021-10-19T14:11:51+00:00\",\"dateModified\":\"2021-11-27T17:55:22+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\"},\"wordCount\":297,\"commentCount\":1,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"nature of roots\",\"nature of roots of quadratic equation\",\"quadratic equation\"],\"articleSection\":[\"Quadratic Equation\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\",\"url\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\",\"name\":\"Nature of Roots of Quadratic Equation - Mathemerize\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-19T14:11:51+00:00\",\"dateModified\":\"2021-11-27T17:55:22+00:00\",\"description\":\"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Nature of Roots of Quadratic Equation\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Nature of Roots of Quadratic Equation - Mathemerize","description":"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/","og_locale":"en_US","og_type":"article","og_title":"Nature of Roots of Quadratic Equation - Mathemerize","og_description":"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.","og_url":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/","og_site_name":"Mathemerize","article_published_time":"2021-10-19T14:11:51+00:00","article_modified_time":"2021-11-27T17:55:22+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Nature of Roots of Quadratic Equation","datePublished":"2021-10-19T14:11:51+00:00","dateModified":"2021-11-27T17:55:22+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/"},"wordCount":297,"commentCount":1,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["nature of roots","nature of roots of quadratic equation","quadratic equation"],"articleSection":["Quadratic Equation"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/","url":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/","name":"Nature of Roots of Quadratic Equation - Mathemerize","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-19T14:11:51+00:00","dateModified":"2021-11-27T17:55:22+00:00","description":"In this post you will learn how to find the nature of roots of quadratic equation using discriminant with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/nature-of-roots-of-quadratic-equation\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Nature of Roots of Quadratic Equation"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6681"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=6681"}],"version-history":[{"count":4,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6681\/revisions"}],"predecessor-version":[{"id":6729,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6681\/revisions\/6729"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6681"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6681"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}