{"id":198,"date":"2022-04-13T10:18:42","date_gmt":"2022-04-13T10:18:42","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=198"},"modified":"2024-08-25T07:45:49","modified_gmt":"2024-08-25T07:45:49","slug":"j-7-solve-a-quadratic-equation-by-completing-the-square","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/j-7-solve-a-quadratic-equation-by-completing-the-square\/","title":{"rendered":"J.7 Solve a quadratic equation by completing the square"},"content":{"rendered":"\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<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Solve a quadratic equation by completing the square<\/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<p class=\"has-text-color has-link-color wp-elements-ec6b65d20e62ebbb451ac1424d199fff\" style=\"color:#f77f42\"><strong>Understanding Completing the Square<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Completing the square is a technique used to rewrite a quadratic equation in the form (x + h)\u00b2 = k, where h and k are constants.<\/li>\n\n\n\n<li>This form makes it easier to solve for x by taking the square root of both sides.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-b252e1e7940587a46b5446a83c4cca71\" style=\"color:#7beb86\"><strong>Steps to Complete the Square<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Ensure the Leading Coefficient is 1:<\/strong> If the leading coefficient (a) is not 1, divide the entire equation by a to make it 1.<\/li>\n\n\n\n<li><strong>Isolate the Quadratic and Linear Terms:<\/strong> Move the constant term (c) to the other side of the equation.<\/li>\n\n\n\n<li><strong>Add the Square of Half the Linear Coefficient:<\/strong> Add (b\/2)\u00b2 to both sides of the equation. This will create a perfect square trinomial on the left side.<\/li>\n\n\n\n<li><strong>Factor the Perfect Square Trinomial:<\/strong> The left side of the equation should now be in the form (x + h)\u00b2.<\/li>\n\n\n\n<li><strong>Solve for x:<\/strong> Take the square root of both sides and solve for x.<\/li>\n<\/ol>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-16d79207b489b19a828d2e9774daa207\" style=\"color:#abe46a\"><strong>Example<\/strong><\/p>\n\n\n\n<p>Solve: x\u00b2 &#8211; 6x + 2 = 0<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Step 1: The leading coefficient is already 1.<\/li>\n\n\n\n<li>Step 2: Move the constant term: x\u00b2 &#8211; 6x = -2<\/li>\n\n\n\n<li>Step 3: Add the square of half the linear coefficient: x\u00b2 &#8211; 6x + 9 = -2 + 9<\/li>\n\n\n\n<li>Step 4: Factor the perfect square trinomial: (x &#8211; 3)\u00b2 = 7<\/li>\n\n\n\n<li>Step 5: Take the square root: x &#8211; 3 = \u00b1\u221a7\n<ul class=\"wp-block-list\">\n<li>x = 3 \u00b1 \u221a7<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-65e42ccaf68d5c3abc795bd855cbb0b3\" style=\"color:#0a5d21\"><strong>Key Points<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Completing the square is a useful technique for solving quadratic equations, especially when factoring is not straightforward.<\/li>\n\n\n\n<li>The goal is to create a perfect square trinomial on the left side of the equation.<\/li>\n\n\n\n<li>Adding the square of half the linear coefficient is the key step in completing the square.<\/li>\n\n\n\n<li>Once the equation is in the form (x + h)\u00b2 = k, solving for x is straightforward by taking the square root.<\/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:#acc8f2\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Solve by completing the square.<\/strong><\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 22<em>s<\/em>&nbsp;= 1<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-b12a3d2c965e072179a28c87fa79a5e2\" style=\"color:#b00012\"><br>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 class=\"has-large-font-size\"><em>s<\/em>&nbsp;=&nbsp;________&nbsp;or&nbsp;<em>s<\/em>&nbsp;=&nbsp;________<\/p>\n<\/div><\/div>\n\n\n\n<p><strong>Step 1: Make sure that the left side of the equation looks like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>.<\/strong><\/p>\n\n\n\n<p>This step does not apply, because the left side of the equation already looks like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>.<\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 22<em>s<\/em>&nbsp;=&nbsp;1<\/p>\n\n\n\n<p>Step 2: Add (<em>b<\/em>\/2)<sup>2<\/sup>&nbsp;to both sides.<\/p>\n\n\n\n<p>Since <em>b<\/em>=22,(b\/2)<sup>2<\/sup> =(22\/2)<sup>2<\/sup> =11<sup>2<\/sup> =121.Add 121 to both sides.<\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 22<em>s<\/em>&nbsp;+ 121 = 122.<\/p>\n\n\n\n<p>Step 3: Factorise the left side as (<em>x<\/em>&nbsp;+&nbsp;<em>b<\/em>\/2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>In general, an expression of the form <em>x<\/em><sup>2<\/sup>&nbsp;&nbsp;+&nbsp;<em>bx<\/em>&nbsp;+(&nbsp;<em>b<\/em>\/2)<sup>2<\/sup>&nbsp;can be factorised as (<em>x<\/em>&nbsp;+&nbsp;<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>The expression <em>s<\/em><sup>2<\/sup>&nbsp;+ 22<em>s<\/em>&nbsp;+ 121 is of this form,with <em>b<\/em>=22. So, it can be factorised as (<em>s<\/em>&nbsp;+ 11)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Rewrite the equation with the left side factorised.<\/p>\n\n\n\n<p>(<em>s<\/em>&nbsp;+ 11)<sup>2<\/sup>&nbsp;= 122<\/p>\n\n\n\n<p><strong>Step 4: Take the square root and solve.<\/strong><\/p>\n\n\n\n<p><em>s<\/em>&nbsp;+ 11&nbsp;\u2248&nbsp;\u00b111.05       Take the square <em>root<\/em><br><em>s<\/em>&nbsp;\u2248&nbsp;-11 \u00b1 11.05             Subtract11frombothsides<br><em>s<\/em>&nbsp;\u2248&nbsp;-11 + 11.05&nbsp;&nbsp;or&nbsp;&nbsp;<em>s<\/em>&nbsp;\u2248&nbsp;-11 \u2212 11.05       <em>Split \u00b1 into + or &#8211;<\/em><br><em>s<\/em>&nbsp;\u2248&nbsp;0.05&nbsp;&nbsp;or&nbsp;&nbsp;<em>s<\/em>&nbsp;\u2248&nbsp;-22.05       <em>Simplify<\/em><\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#f3a0f3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Solve by completing the square.<\/strong><\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 6<em>s<\/em>&nbsp;\u2212 11 = 0<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-b12a3d2c965e072179a28c87fa79a5e2\" style=\"color:#b00012\"><br>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><em>s<\/em>&nbsp;=&nbsp;________or&nbsp;<em>s<\/em>&nbsp;=________<\/p>\n<\/div><\/div>\n\n\n\n<p><strong>Step 1: Make sure that the left side of the equation looks like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>.<\/strong><\/p>\n\n\n\n<p>To make the left side of the equation look like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>, add 11 to both sides.<\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 6<em>s<\/em>&nbsp;\u2212 11&nbsp;=&nbsp;0<br><em>s<\/em><sup>2<\/sup>&nbsp;+ 6<em>s<\/em>&nbsp;=&nbsp;11<\/p>\n\n\n\n<p>Step 2: Add (<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;to both sides.<\/p>\n\n\n\n<p>Since <em>b<\/em>=6, (<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;(6\/2)<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;3<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;9.Add 9 to both sides.<\/p>\n\n\n\n<p><em>s<\/em><sup>2<\/sup>&nbsp;+ 6<em>s<\/em>&nbsp;+ 9 = 20<\/p>\n\n\n\n<p>Step 3: Factorise the left side as (<em>x<\/em>&nbsp;+&nbsp;<em>b<\/em>\/2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>In general, an expression of the form <em>x<\/em><sup>2<\/sup>&nbsp;&nbsp;+&nbsp;<em>bx<\/em>&nbsp;+&nbsp;(<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;can be factorised as (<em>x<\/em>&nbsp;+&nbsp;<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>The expression <em>s<\/em><sup>2<\/sup>&nbsp;+ 6<em>s<\/em>&nbsp;+ 9 is of this form,with <em>b<\/em>=6.So,it can be  factorised as (<em>s<\/em>&nbsp;+ 3)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Rewrite the equation with the left side factorised.<\/p>\n\n\n\n<p>(<em>s<\/em>&nbsp;+ 3)<sup>2<\/sup>&nbsp;= 20<\/p>\n\n\n\n<p><strong>Step 4: Take the square root and solve.<\/strong><\/p>\n\n\n\n<p><em>s<\/em>&nbsp;+ 3&nbsp;\u2248&nbsp;\u00b14.47                                <em>Take the square root<\/em><br><em>s<\/em>&nbsp;\u2248&nbsp;-3 \u00b1 4.47               Subtract3 from  both   sides<br><em>s<\/em>&nbsp;\u2248&nbsp;-3 + 4.47&nbsp;&nbsp;or&nbsp;&nbsp;<em>s<\/em>&nbsp;\u2248&nbsp;-3 \u2212 4.47    <em>Split \u00b1 into + or &#8211;<\/em><br><em>s<\/em>&nbsp;\u2248&nbsp;1.47&nbsp;&nbsp;or&nbsp;&nbsp;<em>s<\/em>&nbsp;\u2248&nbsp;-7.47                     <em>Simplify<\/em><\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background has-large-font-size\" style=\"background-color:#9cf1a9\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p><strong>Solve by completing the square.<\/strong><\/p>\n\n\n\n<p class=\"has-primary-color has-text-color has-link-color wp-elements-8fddc3f418a88d785475d7baca9951ed\"><em>q<\/em><sup>2<\/sup>&nbsp;+ 22<em>q<\/em>&nbsp;= 13<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-b12a3d2c965e072179a28c87fa79a5e2\" style=\"color:#b00012\"><br>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><em>q<\/em>&nbsp;=&nbsp;________or&nbsp;<em>q<\/em>&nbsp;=&nbsp;________<\/p>\n<\/div><\/div>\n\n\n\n<p><strong>Step 1: Make sure that the left side of the equation looks like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>.<\/strong><\/p>\n\n\n\n<p>This step does not apply, because the left side of the equation already looks like&nbsp;<em>x<\/em><sup>2<\/sup>&nbsp;+&nbsp;<em>b<\/em><em>x<\/em>.<\/p>\n\n\n\n<p><em>q<\/em><sup>2<\/sup>&nbsp;+ 22<em>q<\/em>&nbsp;=&nbsp;13<\/p>\n\n\n\n<p>Step 2: Add (<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;to both sides.<\/p>\n\n\n\n<p>Since <em>b<\/em>=22, (<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;(22\/2)<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;11<sup>2<\/sup>&nbsp;&nbsp;=&nbsp;121.Add 121 to both sides.<\/p>\n\n\n\n<p><em>q<\/em><sup>2<\/sup>&nbsp;+ 22<em>q<\/em>&nbsp;+ 121 = 134<\/p>\n\n\n\n<p>Step 3: Factorise the left side as (<em>x<\/em>&nbsp;+&nbsp;<em>b<\/em>\/2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>In general, an expression of the form <em>x<\/em><sup>2<\/sup>&nbsp;&nbsp;+&nbsp;<em>bx<\/em>&nbsp;+(&nbsp;<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;can be factorised as <em>x<\/em>&nbsp;+&nbsp;(<em>b\/<\/em>2)<sup>2<\/sup>&nbsp;.<\/p>\n\n\n\n<p>The expression q<sup>2<\/sup>+ 22<em>q<\/em>&nbsp;+ 121 is of this form, with <em>b<\/em>=22. So,it can be factorised as (<em>q<\/em>&nbsp;+ 11)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Rewrite the equation with the left side factorised.<\/p>\n\n\n\n<p>(<em>q<\/em>&nbsp;+ 11)<sup>2<\/sup>&nbsp;= 134<\/p>\n\n\n\n<p><strong>Step 4: Take the square root and solve.<\/strong><\/p>\n\n\n\n<p>q&nbsp;+ 11&nbsp;\u2248&nbsp;\u00b111.58                      Take the square root<br><em>q<\/em>&nbsp;\u2248&nbsp;-11 \u00b1 11.58              Subtract 11from both sides<br><em>q<\/em>&nbsp;\u2248&nbsp;-11 + 11.58&nbsp;&nbsp;or&nbsp;&nbsp;<em>q<\/em>&nbsp;\u2248&nbsp;-11 \u2212 11.58     <em>Split \u00b1 into + or &#8211;<\/em><br><em>q<\/em>&nbsp;\u2248&nbsp;0.58&nbsp;&nbsp;or&nbsp;&nbsp;<em>q<\/em>&nbsp;\u2248&nbsp;-22.58                     Simplify<\/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\/76704\/736\/390\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-54.png\" alt=\"\" class=\"wp-image-6956\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-54.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-54-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-54-150x150.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\/76721\/084\/100\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-72.png\" alt=\"\" class=\"wp-image-6958\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-72.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-72-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-72-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n<\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Solve a quadratic equation by completing the square Key Notes : Understanding Completing the Square Steps to Complete the Square Example Solve: x\u00b2 &#8211; 6x + 2 = 0 Key Points Learn with an example Solve by completing the square. s2&nbsp;+ 22s&nbsp;= 1 Write your answers as integers, proper or improper fractions in simplest form,<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/j-7-solve-a-quadratic-equation-by-completing-the-square\/\">Continue reading <span class=\"screen-reader-text\">&#8220;J.7 Solve a quadratic equation by completing the square&#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-198","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/198","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=198"}],"version-history":[{"count":20,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/198\/revisions"}],"predecessor-version":[{"id":13676,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/198\/revisions\/13676"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=198"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}