{"id":495,"date":"2022-04-13T05:21:47","date_gmt":"2022-04-13T05:21:47","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=495"},"modified":"2025-02-23T06:44:05","modified_gmt":"2025-02-23T06:44:05","slug":"y-3-factorise-quadratics-with-leading-coefficient-1","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/y-3-factorise-quadratics-with-leading-coefficient-1\/","title":{"rendered":"Y.3 Factorise quadratics with leading coefficient 1"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong>Factorise quadratics with leading coefficient1<\/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-dc2aa747ff5987681cff94e16e956a90\" 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: x\u00b2 + bx + c, where the leading coefficient (coefficient of x\u00b2) is 1.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-30e75a98dd53fbddb2ba888d46123119\" style=\"color:#000060\"><strong>Factorisation as Reverse Expansion<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Factorising means writing the quadratic expression as a product of two binomials.<\/li>\n\n\n\n<li>Example: x\u00b2 + 5x + 6 = (x + 2)(x + 3).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-c32c2056fe074832f98fd46a64155d22\" style=\"color:#000060\"><strong>Finding Two Numbers<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">Identify two numbers that:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Multiply to give c (the constant term).<\/li>\n\n\n\n<li>Add to give b (the coefficient of x).<\/li>\n\n\n\n<li>Example: In x\u00b2 + 7x + 12, find two numbers that multiply to 12 and add to 7 \u2192 (3 and 4).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-c40d82630cc7fad489dead6f998248e4\" style=\"color:#000060\"><strong>Writing the Factors<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Once the numbers are found, express the quadratic as:<\/li>\n\n\n\n<li>x\u00b2 + bx + c = (x + p)(x + q)<\/li>\n\n\n\n<li>Example: x\u00b2 + 7x + 12 = (x + 3)(x + 4).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-f7c4a402599a68faf4300120638d86c3\" style=\"color:#000060\"><strong>Checking the Factorisation<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">Expand the binomials to verify:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>(x + 3)(x + 4) = x\u00b2 + 4x + 3x + 12 = x\u00b2 + 7x + 12 \u2714<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-9fb9620f7199f8f2abf371e7dcf876f3\" style=\"color:#000060\"><strong>Special Cases<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\">Perfect Square Quadratics: Example:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>x\u00b2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)\u00b2<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">Difference of Squares: Example:<\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>x\u00b2 &#8211; 9 = (x + 3)(x &#8211; 3).<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">To&nbsp;factorise a quadratic of the form&nbsp;ax<sup>2<\/sup> + bx + c,&nbsp;write it&nbsp;as(x + r1) (x + r2)<\/p>\n\n\n\n<p class=\"has-large-font-size\">where&nbsp;c = r1 . r2&nbsp;and&nbsp;b = r1 + r2.<\/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:#f2dbaa\"><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-cb815b9eb58c5ce870c00c0cc8d58411\" style=\"color:#b00012\">\ud83d\udc49 <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>s<sup>2<\/sup>+5s+4=&#8212;&#8211;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>s<sup>2<\/sup>+5s+4<\/p>\n\n\n\n<p>The&nbsp;c&nbsp;term is&nbsp;4,&nbsp;so you need to find a pair of factors with a product of&nbsp;4.&nbsp;The&nbsp;b&nbsp;term is&nbsp;5,&nbsp;so you need to find a pair of factors with a sum of&nbsp;5.&nbsp;Since the product is positive&nbsp;(4)&nbsp;and the sum is positive&nbsp;(5),&nbsp;you need both factors to be&nbsp;positive.<\/p>\n\n\n\n<p>Make&nbsp;a list of the possible factor pairs with a product of&nbsp;4,&nbsp;and then find the one with a sum of&nbsp;5.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675749418554_925\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;c=4<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td><strong>1 . <\/strong><strong>4<\/strong><strong>=<\/strong><strong>4<\/strong><\/td><td><strong>1<\/strong>+<strong>4<\/strong><strong>=<\/strong><strong>5<\/strong><\/td><\/tr><tr><td>2 . 2=4<\/td><td>2+2=4<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;1&nbsp;and&nbsp;4&nbsp;have a sum of&nbsp;5.&nbsp;Use those numbers to factorise&nbsp;s2+5s+4.<\/p>\n\n\n\n<p>s<sup>2<\/sup>+5s+4<\/p>\n\n\n\n<p>(s+<strong>1<\/strong>)(s+<strong>4<\/strong>)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(s+1)(s+4)<\/p>\n\n\n\n<p>s<sup>2<\/sup>+s+4s+4Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>s<sup>2<\/sup>+5s+4<\/p>\n\n\n\n<p>Yes,&nbsp;s<sup>2<\/sup>+5s+4=(s+1)(s+4).<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#e8f8a9\"><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-cb815b9eb58c5ce870c00c0cc8d58411\" style=\"color:#b00012\">\ud83d\udc49 <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>f<sup>2<\/sup>+6f+8=&#8212;&#8211;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>f2+6f+8<\/p>\n\n\n\n<p>The&nbsp;c&nbsp;term is&nbsp;8,&nbsp;so you need to find a pair of factors with a product of&nbsp;8.&nbsp;The&nbsp;b&nbsp;term is&nbsp;6,&nbsp;so you need to find a pair of factors with a sum of&nbsp;6.&nbsp;Since the product is positive&nbsp;(8)&nbsp;and the sum is positive&nbsp;(6),&nbsp;you need both factors to be&nbsp;positive.<\/p>\n\n\n\n<p>Make&nbsp;a list of the possible factor pairs with a product of&nbsp;8,&nbsp;and then find the one with a sum of&nbsp;6.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675749793975_926\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;c=8<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td>1 . 8=8<\/td><td>1+8=9<\/td><\/tr><tr><td><strong>2 . <\/strong><strong>4<\/strong><strong>=<\/strong><strong>8<\/strong><\/td><td><strong>2<\/strong>+<strong>4<\/strong><strong>=<\/strong><strong>6<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;2&nbsp;and&nbsp;4&nbsp;have a sum of&nbsp;6.&nbsp;Use those numbers to factorise&nbsp;f<sup>2<\/sup>+6f+8.<\/p>\n\n\n\n<p>f<sup>2<\/sup>+6f+8<\/p>\n\n\n\n<p>(f+<strong>2<\/strong>)(f+<strong>4<\/strong>)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(f+2)(f+4)<\/p>\n\n\n\n<p>f2+2f+4f+8Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>f2+6f+8<\/p>\n\n\n\n<p>Yes,&nbsp;f<sup>2<\/sup>+6f+8=(f+2)(f+4).<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#9ded9a\"><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-cb815b9eb58c5ce870c00c0cc8d58411\" style=\"color:#b00012\">\ud83d\udc49 <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>x<sup>2<\/sup>+10x+9=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>x<sup>2<\/sup>+10x+9<\/p>\n\n\n\n<p>The&nbsp;c&nbsp;term is&nbsp;9,&nbsp;so you need to find a pair of factors with a product of&nbsp;9.&nbsp;The&nbsp;b&nbsp;term is&nbsp;10,&nbsp;so you need to find a pair of factors with a sum of&nbsp;10.&nbsp;Since the product is positive&nbsp;(9)&nbsp;and the sum is positive&nbsp;(10),&nbsp;you need both factors to be&nbsp;positive.<\/p>\n\n\n\n<p>Make&nbsp;a list of the possible factor pairs with a product of&nbsp;9,&nbsp;and then find the one with a sum of&nbsp;10.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675750055078_768\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;c=9<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td><strong>1 . <\/strong><strong>9<\/strong><strong>=<\/strong><strong>9<\/strong><\/td><td><strong>1<\/strong>+<strong>9<\/strong><strong>=<\/strong><strong>10<\/strong><\/td><\/tr><tr><td>3 . 3=9<\/td><td>3+3=6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;1&nbsp;and&nbsp;9&nbsp;have a sum of&nbsp;10.&nbsp;Use those numbers to factorise&nbsp;x<sup>2<\/sup>+10x+9.<\/p>\n\n\n\n<p>x<sup>2<\/sup>+10x+9<\/p>\n\n\n\n<p>(x+<strong>1<\/strong>)(x+<strong>9<\/strong>)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(x+1)(x+9)<\/p>\n\n\n\n<p>x<sup>2<\/sup>+x+9x+9      Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>x<sup>2<\/sup>+10x+9<\/p>\n\n\n\n<p>Yes,&nbsp;x<sup>2<\/sup>+10x+9=(x+1)(x+9).<\/p>\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\/86516\/865\/296\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-169.png\" alt=\"\" class=\"wp-image-9036\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-169.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-169-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-169-150x150.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\/86175\/124\/524\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-174.png\" alt=\"\" class=\"wp-image-9037\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-174.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-174-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-174-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Factorise quadratics with leading coefficient1 Key Notes: Understanding Quadratic Expressions Factorisation as Reverse Expansion Finding Two Numbers Identify two numbers that: Writing the Factors Checking the Factorisation Expand the binomials to verify: Special Cases Perfect Square Quadratics: Example: Difference of Squares: Example: To&nbsp;factorise a quadratic of the form&nbsp;ax2 + bx + c,&nbsp;write it&nbsp;as(x + r1)<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/y-3-factorise-quadratics-with-leading-coefficient-1\/\">Continue reading <span class=\"screen-reader-text\">&#8220;Y.3 Factorise quadratics with leading coefficient 1&#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-495","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\/495","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=495"}],"version-history":[{"count":19,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/495\/revisions"}],"predecessor-version":[{"id":19942,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/495\/revisions\/19942"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}