{"id":156144,"date":"2022-03-26T01:25:51","date_gmt":"2022-03-25T19:55:51","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/discriminant-explanation-formula-and-relationship-between-roots-and-discriminant\/"},"modified":"2024-12-13T17:57:08","modified_gmt":"2024-12-13T12:27:08","slug":"discriminant-explanation-formula-and-relationship-between-roots-and-discriminant","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/","title":{"rendered":"Discriminant \u2013 Explanation, Formula and Relationship between Roots"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/#What_is_Discriminant\" title=\"What is Discriminant?\">What is Discriminant?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/#Formula_and_Relationship_between_Roots_and_Discriminant\" title=\"Formula and Relationship between Roots and Discriminant\">Formula and Relationship between Roots and Discriminant<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/#Things_to_Remember_While_Using_Quadratic_Formula\" title=\"Things to Remember While Using Quadratic Formula\">Things to Remember While Using Quadratic Formula<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/#Formula_and_Relationship_between_Roots_and_Discriminant-2\" title=\"Formula and Relationship between Roots and Discriminant\">Formula and Relationship between Roots and Discriminant<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Discriminant\"><\/span>What is Discriminant?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Discriminant \u2013 Explanation: A <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/discriminant\/\">discriminant<\/a> is a mathematical expression that is used to determine the nature of a particular type of mathematical equation. In most cases, the discriminant is used to determine whether a particular equation has two or three solutions. For equations that have three solutions, the discriminant will be zero. For equations with two solutions, the discriminant will be a positive number.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156143 size-medium\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/discriminant-explanation-formula-and-relationship-between-roots-and-discriminant-300x212.jpg\" alt=\"Discriminant \u2013 Explanation, Formula and Relationship between Roots and Discriminant\" width=\"300\" height=\"212\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/discriminant-explanation-formula-and-relationship-between-roots-and-discriminant-300x212.jpg?v=1648238147 300w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/discriminant-explanation-formula-and-relationship-between-roots-and-discriminant.jpg?v=1648238147 606w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Formula_and_Relationship_between_Roots_and_Discriminant\"><\/span>Formula and Relationship between Roots and Discriminant<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A discriminant is a calculation that is used to determine the nature of the roots of a quadratic equation. The discriminant is the result of taking the square root of the coefficient of the x squared term in the equation. The discriminant is then used to determine if the roots of the equation are real or complex. If the discriminant is positive, then the roots are real. If the discriminant is negative, then the roots are complex.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Things_to_Remember_While_Using_Quadratic_Formula\"><\/span>Things to Remember While Using Quadratic Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The Quadratic Formula is a mathematical equation that is used to solve quadratic equations. A <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/nature-of-roots-of-quadratic-equation\/\" target=\"_blank\" rel=\"noopener\">quadratic equation<\/a> is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and also x is the variable. The Quadratic Formula can be used to solve any quadratic equation, regardless of the value of a, b, and c.<\/p>\n<p>The Quadratic Formula is as follows:<\/p>\n<p>x = (-b +- sqrt(b^2 &#8211; 4ac))\/(2a)<\/p>\n<p>To use the Quadratic Formula, you need to first isolate the variable, x, on one side of the equation. To do this, you need to subtract bx from both sides of the equation. This will leave you with ax^2 on one side and also c on the other.<\/p>\n<p>Next, take the square root of both sides of the equation. This will leave you with b^2 on one side and 4ac on the other.<\/p>\n<p>Finally, divide both sides of the equation by 2a. This will leave you with the value of x.<\/p>\n<h3 dir=\"ltr\"><span class=\"ez-toc-section\" id=\"Formula_and_Relationship_between_Roots_and_Discriminant-2\"><\/span>Formula and Relationship between Roots and Discriminant<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p dir=\"ltr\">Any polynomial&#8217;s discriminant (\u0394 or D) is defined in terms of its coefficients. Therefore the discriminant formulas for a cubic equation and a quadratic equation are:<\/p>\n<p dir=\"ltr\">Discriminant formula of a quadratic equation:<\/p>\n<p dir=\"ltr\">ax2  bx + c = 0 is<\/p>\n<p dir=\"ltr\">\u0394 or D = b2 \u2212 4ac<\/p>\n<p dir=\"ltr\">Discriminant formula of a cubic equation:<\/p>\n<p dir=\"ltr\">ax + bx\u00b3 + cx\u00b2 + d = 0 is<\/p>\n<p dir=\"ltr\">\u0394 or D = b2c2 \u2212 4ac3 \u2212 4b3d \u221227a2d2 + 18abcd<\/p>\n<p dir=\"ltr\">Relationship between Roots and Discriminant<\/p>\n<p dir=\"ltr\">The values of x that satisfy the equation are also known as the roots of the quadratic equation ax2 + bx + c = 0.<\/p>\n<p dir=\"ltr\">To find them, use the quadratic formula:<\/p>\n<p dir=\"ltr\">X =<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mi&gt;b&lt;\/mi&gt;&lt;mo&gt;&amp;#x00B1;&lt;\/mo&gt;&lt;msqrt&gt;&lt;mi&gt;D&lt;\/mi&gt;&lt;\/msqrt&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;mi&gt;a&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"mfrac\"><span id=\"MathJax-Span-4\" class=\"mrow\"><span id=\"MathJax-Span-5\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-6\" class=\"mi\">b<\/span><span id=\"MathJax-Span-7\" class=\"mo\">\u00b1<\/span><span id=\"MathJax-Span-8\" class=\"msqrt\"><span id=\"MathJax-Span-9\" class=\"mrow\"><span id=\"MathJax-Span-10\" class=\"mi\">D<\/span><\/span>\u2212\u2212\u221a<\/span><\/span><span id=\"MathJax-Span-11\" class=\"mrow\"><span id=\"MathJax-Span-12\" class=\"mn\">2<\/span><span id=\"MathJax-Span-13\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mi>D<\/mi><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/math>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">Although we cannot discover the roots using the discriminant alone, we can determine the nature of the roots in the following way.<\/p>\n<p dir=\"ltr\">If discriminant is positive:<\/p>\n<p dir=\"ltr\">There are also two real roots to the quadratic equation if<\/p>\n<p dir=\"ltr\">D &gt; 0.<\/p>\n<p dir=\"ltr\">Therefore this is because the roots of D &gt; 0 also are provided by x =<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-2-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mi&gt;b&lt;\/mi&gt;&lt;mo&gt;&amp;#x00B1;&lt;\/mo&gt;&lt;msqrt&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;Positive number&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/msqrt&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;mi&gt;a&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-14\" class=\"math\"><span id=\"MathJax-Span-15\" class=\"mrow\"><span id=\"MathJax-Span-16\" class=\"mfrac\"><span id=\"MathJax-Span-17\" class=\"mrow\"><span id=\"MathJax-Span-18\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-19\" class=\"mi\">b<\/span><span id=\"MathJax-Span-20\" class=\"mo\">\u00b1<\/span><span id=\"MathJax-Span-21\" class=\"msqrt\"><span id=\"MathJax-Span-22\" class=\"mrow\"><span id=\"MathJax-Span-23\" class=\"texatom\"><span id=\"MathJax-Span-24\" class=\"mrow\"><span id=\"MathJax-Span-25\" class=\"mtext\">Positive number<\/span><\/span><\/span><\/span>\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u221a<\/span><\/span><span id=\"MathJax-Span-26\" class=\"mrow\"><span id=\"MathJax-Span-27\" class=\"mn\">2<\/span><span id=\"MathJax-Span-28\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow class=\"MJX-TeXAtom-ORD\"><mtext>Positive number<\/mtext><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/math>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">And a real number is always the square root of a positive number.<\/p>\n<p dir=\"ltr\">When the discriminant of a quadratic equation exceeds 0, it has two separate and real-number roots.<\/p>\n<p dir=\"ltr\">If discriminant is negative:<\/p>\n<p dir=\"ltr\">The quadratic equation has two different complex roots if<\/p>\n<p dir=\"ltr\">D &lt; 0.<\/p>\n<p dir=\"ltr\">This is because the roots of D &lt; 0 are provided by x =<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-3-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mi&gt;b&lt;\/mi&gt;&lt;mo&gt;&amp;#x00B1;&lt;\/mo&gt;&lt;msqrt&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;Negative number&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;\/msqrt&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;mi&gt;a&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-29\" class=\"math\"><span id=\"MathJax-Span-30\" class=\"mrow\"><span id=\"MathJax-Span-31\" class=\"mfrac\"><span id=\"MathJax-Span-32\" class=\"mrow\"><span id=\"MathJax-Span-33\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-34\" class=\"mi\">b<\/span><span id=\"MathJax-Span-35\" class=\"mo\">\u00b1<\/span><span id=\"MathJax-Span-36\" class=\"msqrt\"><span id=\"MathJax-Span-37\" class=\"mrow\"><span id=\"MathJax-Span-38\" class=\"texatom\"><span id=\"MathJax-Span-39\" class=\"mrow\"><span id=\"MathJax-Span-40\" class=\"mtext\">Negative number<\/span><\/span><\/span><\/span>\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u2212\u221a<\/span><\/span><span id=\"MathJax-Span-41\" class=\"mrow\"><span id=\"MathJax-Span-42\" class=\"mn\">2<\/span><span id=\"MathJax-Span-43\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow class=\"MJX-TeXAtom-ORD\"><mtext>Negative number<\/mtext><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/math>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">and so when you take the square root of a negative number, you always get an imaginary number.<\/p>\n<p dir=\"ltr\">If discriminant is equal to zero:<\/p>\n<p dir=\"ltr\">Therefore the quadratic equation has two equal real roots if D = 0.<\/p>\n<p dir=\"ltr\">This is because the roots of D = 0 are provided by x =<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mi&gt;b&lt;\/mi&gt;&lt;mo&gt;&amp;#x00B1;&lt;\/mo&gt;&lt;msqrt&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/msqrt&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;mi&gt;a&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-44\" class=\"math\"><span id=\"MathJax-Span-45\" class=\"mrow\"><span id=\"MathJax-Span-46\" class=\"mfrac\"><span id=\"MathJax-Span-47\" class=\"mrow\"><span id=\"MathJax-Span-48\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-49\" class=\"mi\">b<\/span><span id=\"MathJax-Span-50\" class=\"mo\">\u00b1<\/span><span id=\"MathJax-Span-51\" class=\"msqrt\"><span id=\"MathJax-Span-52\" class=\"mrow\"><span id=\"MathJax-Span-53\" class=\"mn\">0<\/span><\/span>\u2013\u221a<\/span><\/span><span id=\"MathJax-Span-54\" class=\"mrow\"><span id=\"MathJax-Span-55\" class=\"mn\">2<\/span><span id=\"MathJax-Span-56\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mn>0<\/mn><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/math>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">and 0 would be the square root. The equation thus becomes x = \u2212b\/2a, which is a single number. When a quadratic equation&#8217;s discriminant is 0, it has only one real root.<\/p>\n<p dir=\"ltr\">For example, the given quadratic equation is \u2013<\/p>\n<p dir=\"ltr\">6&#215;2 + 10x \u2013 1 = 0<\/p>\n<p dir=\"ltr\">From the above equation, it can be seen that:<\/p>\n<p dir=\"ltr\">a = 6,<\/p>\n<p dir=\"ltr\">b = 10,<\/p>\n<p dir=\"ltr\">also c = \u22121<\/p>\n<p dir=\"ltr\">However Applying the numbers in discriminant \u2013<\/p>\n<p dir=\"ltr\">b2 \u2212 4ac<\/p>\n<p dir=\"ltr\">= 102 \u2013 4 (6) (\u22121) =100 + 24<\/p>\n<p dir=\"ltr\">= 124<\/p>\n<p dir=\"ltr\">Given that, the discriminant amounts to be a positive number, there are two solutions to the quadratic equation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is Discriminant? 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