{"id":499,"date":"2022-04-13T05:22:17","date_gmt":"2022-04-13T05:22:17","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=499"},"modified":"2025-02-23T06:50:15","modified_gmt":"2025-02-23T06:50:15","slug":"y-5-factorise-quadratics-special-cases","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/y-5-factorise-quadratics-special-cases\/","title":{"rendered":"Y.5 Factorise quadratics: special cases"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Factorise quadratics: special cases<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-4d4696560822e7cb72ec4be90d4370b1\" style=\"color:#74008b\">Key Notes:<\/p>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-3ccab5f1cb3106917291f01e96c8ed5b\" style=\"color:#000060\"><strong>Understanding Quadratic Expressions<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>A quadratic expression is in the form <strong>ax\u00b2 + bx + c<\/strong>.<\/li>\n\n\n\n<li>Special cases involve patterns that make factorisation easier.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-b440eb800e382580e993955a24372f89\" style=\"color:#000060\"><strong>Perfect Square Trinomials<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>These take the form <strong>a\u00b2 + 2ab + b\u00b2 = (a + b)\u00b2<\/strong> or <strong>a\u00b2 &#8211; 2ab + b\u00b2 = (a &#8211; b)\u00b2<\/strong>.<\/li>\n\n\n\n<li>Example: <strong>x\u00b2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)\u00b2<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-cdf82e9b433fcfd394f11b033aa3bd9f\" style=\"color:#000060\"><strong>Difference of Squares<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>This follows the identity <strong>a\u00b2 &#8211; b\u00b2 = (a &#8211; b)(a + b)<\/strong>.<\/li>\n\n\n\n<li>Example: <strong>x\u00b2 &#8211; 9 = (x &#8211; 3)(x + 3)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-ead8daf15f4e7acac1b5a2c27ed33ce8\" style=\"color:#000060\"><strong>Common Factor Method<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Always check for common factors before applying special cases.<\/li>\n\n\n\n<li>Example: <strong>4x\u00b2 &#8211; 36 = 4(x\u00b2 &#8211; 9) = 4(x &#8211; 3)(x + 3)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-f34729eda6f597ff15af9241660f324c\" style=\"color:#000060\"><strong>Zero Product Property<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>If a quadratic expression is factorised and set to zero, solve by setting each factor to zero.<\/li>\n\n\n\n<li>Example: <strong>(x &#8211; 5)(x + 2) = 0<\/strong> gives <strong>x = 5 or x = -2<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-4efd3ac2a3a9a0f58dd03d9000e0c95e\" style=\"color:#000060\"><strong>Recognizing Special Cases<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Identify perfect squares and differences of squares quickly.<\/li>\n\n\n\n<li>Look for missing middle terms (b = 0) in the difference of squares.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-454d264368ef20f1e35bfdf76b79f2eb\" style=\"color:#000060\"><strong>Practice with Different Forms<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Example 1: <strong>x\u00b2 + 16x + 64 = (x + 8)\u00b2<\/strong>.<\/li>\n\n\n\n<li>Example 2: <strong>49y\u00b2 &#8211; 25 = (7y &#8211; 5)(7y + 5)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-04ab9f6c97c68a3cf237468d32b597f1\" style=\"color:#000060\"><strong>Application in Problem Solving<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Use factorisation to simplify algebraic expressions.<\/li>\n\n\n\n<li>Apply in solving quadratic equations efficiently.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">Factorising&nbsp;perfect square&nbsp;trinomials:<\/p>\n\n\n\n<p class=\"has-large-font-size\">a<sup>2<\/sup>+2ab+b<sup>2<\/sup>=(a+b)<sup>2<\/sup><\/p>\n\n\n\n<p class=\"has-large-font-size\">a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>=(a\u2013b)<sup>2<\/sup> <\/p>\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\" style=\"background-color:#d5f3cc\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\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<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 class=\"has-text-color has-link-color wp-elements-19cdee1d047d0cef7db099807c0340e4\" style=\"color:#b00012\">\ud83e\udd4e<strong> Factorise.<\/strong><\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20138c+16=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Notice&nbsp;that&nbsp;c<sup>2<\/sup>\u20138c+16&nbsp;is a perfect square trinomial because it can be written in the form&nbsp;a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>,&nbsp;where&nbsp;a&nbsp;is&nbsp;c&nbsp;and&nbsp;b&nbsp;is&nbsp;<strong>4<\/strong>.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup><\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20132c<strong>4<\/strong>+<strong>4<\/strong><sup>2<\/sup><\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20138c+16<\/p>\n\n\n\n<p>Now&nbsp;use the formula for factorising perfect square&nbsp;trinomials.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>=(a\u2013b)<sup>2<\/sup><\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20132c<strong>4<\/strong>+<strong>4<\/strong><sup>2<\/sup>=(c\u2013<strong>4<\/strong>)<sup>2<\/sup><\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20138c+16=(c\u20134)<sup>2<\/sup><\/p>\n\n\n\n<p>The&nbsp;factorised form of&nbsp;c<sup>2<\/sup>\u20138c+16&nbsp;is&nbsp;(c\u20134)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(c\u20134)<sup>2<\/sup><\/p>\n\n\n\n<p>(c\u20134)(c\u20134)Expand<\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20134c\u20134c+16Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>c<sup>2<\/sup>\u20138c+16<\/p>\n\n\n\n<p>Yes,&nbsp;c<sup>2<\/sup>\u20138c+16=(c\u20134)<sup>2<\/sup>.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#e4eaa3\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\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<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 class=\"has-text-color has-link-color wp-elements-19cdee1d047d0cef7db099807c0340e4\" style=\"color:#b00012\">\ud83e\udd4e<strong> Factorise.<\/strong><\/p>\n\n\n\n<p>9x<sup>2<\/sup>\u20136x+1=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Notice&nbsp;that&nbsp;9x<sup>2<\/sup>\u20136x+1&nbsp;is a perfect square trinomial because it can be written in the form&nbsp;a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>,&nbsp;where&nbsp;a&nbsp;is&nbsp;<strong>3<\/strong>x&nbsp;and&nbsp;b&nbsp;is&nbsp;<strong>1<\/strong>.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup><\/p>\n\n\n\n<p>(<strong>3<\/strong>x)<sup>2<\/sup>\u20132 . <strong>3<\/strong>x . <strong>1<\/strong>+<strong>1<\/strong><sup>2<\/sup><\/p>\n\n\n\n<p>9x<sup>2<\/sup>\u20136x+1<\/p>\n\n\n\n<p>Now&nbsp;use the formula for factorising perfect square&nbsp;trinomials.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u20132ab+b<sup>2<\/sup>=(a\u2013b)<sup>2<\/sup><\/p>\n\n\n\n<p>(<strong>3<\/strong>x)<sup>2<\/sup>\u20132 . <strong>3<\/strong>x . <strong>1<\/strong>+<strong>1<\/strong>2=(<strong>3<\/strong>x\u2013<strong>1<\/strong>)<sup>2<\/sup><\/p>\n\n\n\n<p>9x<sup>2<\/sup>\u20136x+1=(3x\u20131)<sup>2<\/sup><\/p>\n\n\n\n<p>The&nbsp;factorised form of&nbsp;9x<sup>2<\/sup>\u20136x+1&nbsp;is&nbsp;(3x\u20131)<sup>2<\/sup>.<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(3x\u20131)<sup>2<\/sup><\/p>\n\n\n\n<p>(3x\u20131)(3x\u20131)Expand<\/p>\n\n\n\n<p>9x<sup>2<\/sup>\u20133x\u20133x+1Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>9x<sup>2<\/sup>\u20136x+1<\/p>\n\n\n\n<p>Yes,&nbsp;9x<sup>2<\/sup>\u20136x+1=(3x\u20131)<sup>2<\/sup>.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#bff0ad\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\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-large-font-size\"><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-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-b70f13e442d773e73218a7cb552c3c10\" style=\"color:#b00012\">\ud83e\udd4e <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>z<sup>2<\/sup>\u20131=&#8212;&#8211;<\/p>\n<\/div><\/div>\n\n\n\n<p>Notice&nbsp;that&nbsp;z<sup>2<\/sup>\u20131&nbsp;is a difference of squares, because it can be written in the form&nbsp;a<sup>2<\/sup>\u2013b<sup>2<\/sup>,&nbsp;where&nbsp;a&nbsp;is&nbsp;z&nbsp;and&nbsp;b&nbsp;is&nbsp;<strong>1<\/strong>.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u2013b<sup>2<\/sup><\/p>\n\n\n\n<p>z<sup>2<\/sup>\u2013<strong>1<\/strong><sup>2<\/sup><\/p>\n\n\n\n<p>z<sup>2<\/sup>\u20131<\/p>\n\n\n\n<p>Now&nbsp;use the formula for factorising a difference of&nbsp;squares.<\/p>\n\n\n\n<p>a<sup>2<\/sup>\u2013b<sup>2<\/sup>=(a+b)(a\u2013b)<\/p>\n\n\n\n<p>z<sup>2<\/sup>\u2013<strong>1<\/strong><sup>2<\/sup>=(z+<strong>1<\/strong>)(z\u2013<strong>1<\/strong>)<\/p>\n\n\n\n<p>z<sup>2<\/sup>\u20131=(z+1)(z\u20131)<\/p>\n\n\n\n<p>The&nbsp;factorised form of&nbsp;z<sup>2<\/sup>\u20131&nbsp;is&nbsp;(z+1)(z\u20131).<\/p>\n\n\n\n<p class=\"has-large-font-size\">Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(z+1)(z\u20131)<\/p>\n\n\n\n<p>z<sup>2<\/sup>+z\u2013z\u20131Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>z<sup>2<\/sup>\u20131<\/p>\n\n\n\n<p>Yes,&nbsp;z<sup>2<\/sup>\u20131=(z+1)(z\u20131).<\/p>\n<\/div><\/div>\n<\/div><\/div>\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\/85392\/314\/459\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-167.png\" alt=\"\" class=\"wp-image-9029\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-167.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-167-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-167-150x150.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\/86237\/907\/741\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-172.png\" alt=\"\" class=\"wp-image-9031\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-172.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-172-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-172-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Factorise quadratics: special cases Key Notes: Understanding Quadratic Expressions Perfect Square Trinomials Difference of Squares Common Factor Method Zero Product Property Recognizing Special Cases Practice with Different Forms Application in Problem Solving Factorising&nbsp;perfect square&nbsp;trinomials: a2+2ab+b2=(a+b)2 a2\u20132ab+b2=(a\u2013b)2 Learn with an example \ud83e\udd4e Factorise. c2\u20138c+16=&#8212;&#8212; Notice&nbsp;that&nbsp;c2\u20138c+16&nbsp;is a perfect square trinomial because it can be written in the<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/y-5-factorise-quadratics-special-cases\/\">Continue reading <span class=\"screen-reader-text\">&#8220;Y.5 Factorise quadratics: special cases&#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-499","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"_links":{"self":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/499","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=499"}],"version-history":[{"count":18,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/499\/revisions"}],"predecessor-version":[{"id":19946,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/499\/revisions\/19946"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}