{"id":200,"date":"2022-04-13T10:18:59","date_gmt":"2022-04-13T10:18:59","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=200"},"modified":"2024-08-25T07:44:38","modified_gmt":"2024-08-25T07:44:38","slug":"j-8-solve-a-quadratic-equation-using-the-quadratic-formula","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/j-8-solve-a-quadratic-equation-using-the-quadratic-formula\/","title":{"rendered":"J.8 Solve a quadratic equation using the quadratic formula"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Solve a quadratic equation using the quadratic formula<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-8550df6181cd5d83aa7a08ef336a4ca1\" style=\"color:#74008b\">Key Notes :<\/p>\n\n\n\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p class=\"has-text-color has-link-color wp-elements-8c89973ca205502de4f1c1ff35da7787\" style=\"color:#4ddf20\"><strong>Understanding the Quadratic Formula<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The quadratic formula is a general formula used to solve any quadratic equation of the form ax\u00b2 + bx + c = 0.<\/li>\n\n\n\n<li>It is derived from completing the square and provides a direct method to find the solutions.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">The&nbsp;<strong>quadratic formula<\/strong>&nbsp;is<\/p>\n\n\n\n<p class=\"has-large-font-size\">x = &#8211; b \u00b1&nbsp;\u221ab<sup>2<\/sup> -4ac \/ 2a<\/p>\n\n\n\n<p>where a, b, and c are the coefficients of the quadratic equation.<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-c902b6408d59ace2796ed845c6cd67b4\" style=\"color:#68b1ff\"><strong>Steps to Solve Using the Quadratic Formula<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identify the Coefficients:<\/strong> Determine the values of a, b, and c from the given quadratic equation.<\/li>\n\n\n\n<li><strong>Substitute into the Formula:<\/strong> Substitute the values of a, b, and c into the quadratic formula.<\/li>\n\n\n\n<li><strong>Simplify and Solve:<\/strong> Evaluate the expression under the square root (discriminant), take the square root, and simplify the resulting expression.<\/li>\n\n\n\n<li><strong>Find the Solutions:<\/strong> Solve for x by considering both the positive and negative square root values.<\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-6dd45570ce37d1132e4e39554fcd1ffe\" style=\"color:#73ea40\"><strong>Example<\/strong><\/p>\n\n\n\n<p>Solve: 2x\u00b2 &#8211; 5x + 3 = 0<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Step 1: a = 2, b = -5, c = 3<\/li>\n\n\n\n<li>Step 2: Substitute into the formula: x = (-(-5) \u00b1 \u221a((-5)\u00b2 &#8211; 4(2)(3))) \/ (2(2))<\/li>\n\n\n\n<li>Step 3: Simplify: x = (5 \u00b1 \u221a1) \/ 4<\/li>\n\n\n\n<li>Step 4: Solve: x = (5 + 1) \/ 4 = 3\/2 or x = (5 &#8211; 1) \/ 4 = 1<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-4340ad751ba41636dbf32ec6eba4fcdf\" style=\"color:#e66309\"><strong>Key Points<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The quadratic formula is a versatile tool for solving quadratic equations.<\/li>\n\n\n\n<li>It works for any quadratic equation, regardless of whether it can be factored or not.<\/li>\n\n\n\n<li>The discriminant (b\u00b2 &#8211; 4ac) determines the nature of the solutions:\n<ul class=\"wp-block-list\">\n<li>Positive: Two distinct real solutions<\/li>\n\n\n\n<li>Zero: One repeated real solution<\/li>\n\n\n\n<li>Negative: Two complex conjugate solutions<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Practice using the quadratic formula to improve your skills and understanding.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f6c5c5\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p><strong>Solve using the quadratic formula.<\/strong><\/p>\n\n\n\n<p>3<em>j<\/em><sup>2<\/sup>&nbsp;+ 5<em>j<\/em>&nbsp;+ 1 = 0<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-e96586d06c246aacfeaef143426e0223\" style=\"color:#b00012\">Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.<\/p>\n\n\n\n<p>j = __________ or j = _____________<\/p>\n<\/div><\/div>\n\n\n\n<p>Use the quadratic formula to solve 3<em>j<\/em><sup>2<\/sup>&nbsp;+ 5<em>j<\/em>&nbsp;+ 1 = 0.<\/p>\n\n\n\n<p><em>j<\/em> = -b\u00b1&nbsp;\u221a<em>b<\/em><sup>2<\/sup>&nbsp;\u2212 4<em>ac<\/em> \/ 2a<\/p>\n\n\n\n<p>j = -5 \u00b1&nbsp;\u221a5<sup>2 <\/sup> -4(3)(1) \/ 2(3)     Plug in&nbsp;a&nbsp;= 3,&nbsp;b&nbsp;= 5, and&nbsp;c&nbsp;= 1<\/p>\n\n\n\n<p>j = -5 \u00b1&nbsp;\u221a25-12 \/ 6   Multiply<\/p>\n\n\n\n<p>j = -5 \u00b1&nbsp;\u221a13 \/ 6  Subtract<\/p>\n\n\n\n<p>j = -5 +\u221a13 \/ 6 or&nbsp;&nbsp; j = -5-\u221a13 \/ 6   Split \u00b1 into + or &#8211;<\/p>\n\n\n\n<p class=\"has-large-font-size\">j &nbsp;\u2248&nbsp;-0.23&nbsp;&nbsp;or&nbsp;&nbsp;<em>j<\/em>&nbsp;\u2248&nbsp;-1.43   Simplify and round to the nearest hundredth <img decoding=\"async\" src=\"\" alt=\"\"><\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#b3e0ae\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p><strong>Solve using the quadratic formula.<\/strong><\/p>\n\n\n\n<p>5<em>u<\/em><sup>2<\/sup>&nbsp;\u2212 5<em>u<\/em>&nbsp;\u2212 4 = 0<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-e96586d06c246aacfeaef143426e0223\" style=\"color:#b00012\">Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.<\/p>\n\n\n\n<p>u = _____________ or u = ______________<\/p>\n<\/div><\/div>\n\n\n\n<p>Use the quadratic formula to solve 5<em>u<\/em><sup>2<\/sup>&nbsp;\u2212 5<em>u<\/em>&nbsp;\u2212 4 = 0.<\/p>\n\n\n\n<p>u = -b\u00b1&nbsp;\u221a<em>b<\/em><sup>2<\/sup>&nbsp;\u2212 4<em>ac<\/em> \/ 2a<\/p>\n\n\n\n<p>j = -(-5) \u00b1&nbsp;\u221a(-5)<sup>2 <\/sup> -4(5)(-4) \/ 2(5)     Plug in&nbsp;a&nbsp;= 5,&nbsp;b&nbsp;= -5, and&nbsp;c&nbsp;= -4<\/p>\n\n\n\n<p>u = 5 \u00b1&nbsp;\u221a25+80 \/ 10   Multiply<\/p>\n\n\n\n<p>j = 5 \u00b1&nbsp;\u221a105 \/ 10  Add<\/p>\n\n\n\n<p>j = 5 +\u221a105 \/ 10 or&nbsp;&nbsp; j = 5-\u221a105 \/ 10   Split \u00b1 into + or &#8211;<\/p>\n\n\n\n<p>u &nbsp;\u2248&nbsp; 1.52&nbsp;&nbsp;or&nbsp;&nbsp;<em>u<\/em>&nbsp;\u2248&nbsp;-0.52     Simplify and round to the nearest hundredth<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#cbc8eb\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p><strong>Solve using the quadratic formula.<\/strong><\/p>\n\n\n\n<p><em>j<\/em><sup>2<\/sup>&nbsp;+ 6<em>j<\/em>&nbsp;+ 9 = 0<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-e96586d06c246aacfeaef143426e0223\" style=\"color:#b00012\">Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.<\/p>\n\n\n\n<p>j = __________ or j = _____________<\/p>\n<\/div><\/div>\n\n\n\n<p>Use the quadratic formula to solve&nbsp;<em>j<\/em><sup>2<\/sup>&nbsp;+ 6<em>j<\/em>&nbsp;+ 9 = 0.<\/p>\n\n\n\n<p><em>j<\/em> = -b\u00b1&nbsp;\u221a<em>b<\/em><sup>2<\/sup>&nbsp;\u2212 4<em>ac<\/em> \/ 2a<\/p>\n\n\n\n<p>j = -6 \u00b1&nbsp;\u221a6<sup>2 <\/sup> -4(1)(9) \/ 2(1)     Plug in&nbsp;a&nbsp;= 1,&nbsp;b&nbsp;= 6, and&nbsp;c&nbsp;= 9<\/p>\n\n\n\n<p>j = -6 \u00b1&nbsp;\u221a36-36 \/ 2  Multiply<\/p>\n\n\n\n<p>j = -6 \u00b1&nbsp;\u221a0 \/ 2  Subtract<\/p>\n\n\n\n<p>j = -6 +\u221a0 \/ 2 or&nbsp;&nbsp; j = -6-\u221a0 \/ 2   Split \u00b1 into + or &#8211;<\/p>\n\n\n\n<p><em>j<\/em> &nbsp;=&nbsp;-3&nbsp;&nbsp;or&nbsp;&nbsp;<em>j<\/em>&nbsp;=&nbsp;-3  Simplify<\/p>\n\n\n\n<p>The two solutions are the same, so they should be written as a single solution  :  <em>j<\/em>&nbsp;=&nbsp;-3.<\/p>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#d90000\">let&#8217;s practice!<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/76706\/117\/676\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-55.png\" alt=\"\" class=\"wp-image-6964\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-55.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-55-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-55-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/76723\/964\/837\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-73.png\" alt=\"\" class=\"wp-image-6966\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-73.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-73-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-73-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n\n\n\n<p><img decoding=\"async\" src=\"\" alt=\"\"><\/p>\n\n\n\n<p class=\"has-large-font-size\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solve a quadratic equation using the quadratic formula Key Notes : Understanding the Quadratic Formula The&nbsp;quadratic formula&nbsp;is x = &#8211; b \u00b1&nbsp;\u221ab2 -4ac \/ 2a where a, b, and c are the coefficients of the quadratic equation. Steps to Solve Using the Quadratic Formula Example Solve: 2x\u00b2 &#8211; 5x + 3 = 0 Key Points<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/j-8-solve-a-quadratic-equation-using-the-quadratic-formula\/\">Continue reading <span class=\"screen-reader-text\">&#8220;J.8 Solve a quadratic equation using the quadratic formula&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-200","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/200","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=200"}],"version-history":[{"count":17,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/200\/revisions"}],"predecessor-version":[{"id":13675,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/200\/revisions\/13675"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}