{"id":497,"date":"2022-04-13T05:22:02","date_gmt":"2022-04-13T05:22:02","guid":{"rendered":"http:\/\/8thclass.deltapublications.in\/?page_id=497"},"modified":"2025-02-23T06:46:51","modified_gmt":"2025-02-23T06:46:51","slug":"y-4-factorise-quadratics-with-other-leading-coefficients","status":"publish","type":"page","link":"https:\/\/8thclass.deltapublications.in\/index.php\/y-4-factorise-quadratics-with-other-leading-coefficients\/","title":{"rendered":"Y.4 Factorise quadratics with other leading coefficients"},"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 other leading coefficients<\/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-c9ebb313e12f0f03151f046d23947e9f\" style=\"color:#000060\"><strong>Understanding Quadratic Equations<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>A quadratic equation is in the form <strong>ax\u00b2 + bx + c<\/strong>, where <strong>a \u2260 1<\/strong> (i.e., the coefficient of x\u00b2 is not 1).<\/li>\n\n\n\n<li>The goal of factorization is to express the quadratic as a product of two binomials.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-fd90b4964f36be7116674c51430cb281\" style=\"color:#000060\"><strong>Factorization Process<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Step 1:<\/strong> Multiply <strong>a<\/strong> and <strong>c<\/strong> (the coefficient of x\u00b2 and the constant term).<\/li>\n\n\n\n<li><strong>Step 2:<\/strong> Find two numbers that multiply to <strong>a \u00d7 c<\/strong> and add to <strong>b<\/strong> (the coefficient of x).<\/li>\n\n\n\n<li><strong>Step 3:<\/strong> Rewrite the middle term using the two numbers found.<\/li>\n\n\n\n<li><strong>Step 4:<\/strong> Group the terms in pairs.<\/li>\n\n\n\n<li><strong>Step 5:<\/strong> Factor out the common term from each group.<\/li>\n\n\n\n<li><strong>Step 6:<\/strong> Write the final factorized form.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\"><strong>Example 1:<\/strong> Factorize <strong>2x\u00b2 + 7x + 3<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Multiply <strong>a \u00d7 c<\/strong>: <strong>2 \u00d7 3 = 6<\/strong><\/li>\n\n\n\n<li>Find two numbers that multiply to <strong>6<\/strong> and add to <strong>7<\/strong> \u2192 <strong>6 and 1<\/strong><\/li>\n\n\n\n<li>Rewrite: <strong>2x\u00b2 + 6x + 1x + 3<\/strong><\/li>\n\n\n\n<li>Group: <strong>(2x\u00b2 + 6x) + (1x + 3)<\/strong><\/li>\n\n\n\n<li>Factor out common terms: <strong>2x(x + 3) + 1(x + 3)<\/strong><\/li>\n\n\n\n<li>Final factorized form: <strong>(2x + 1)(x + 3)<\/strong><\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\"><strong>Example 2:<\/strong> Factorize <strong>3x\u00b2 &#8211; 8x &#8211; 3<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Multiply <strong>a \u00d7 c<\/strong>: <strong>3 \u00d7 (-3) = -9<\/strong><\/li>\n\n\n\n<li>Find two numbers that multiply to <strong>-9<\/strong> and add to <strong>-8<\/strong> \u2192 <strong>-9 and 1<\/strong><\/li>\n\n\n\n<li>Rewrite: <strong>3x\u00b2 &#8211; 9x + 1x &#8211; 3<\/strong><\/li>\n\n\n\n<li>Group: <strong>(3x\u00b2 &#8211; 9x) + (1x &#8211; 3)<\/strong><\/li>\n\n\n\n<li>Factor out common terms: <strong>3x(x &#8211; 3) + 1(x &#8211; 3)<\/strong><\/li>\n\n\n\n<li>Final factorized form: <strong>(3x + 1)(x &#8211; 3)<\/strong><\/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<ul class=\"wp-block-list has-large-font-size\">\n<li><strong>Perfect Square Quadratics:<\/strong> If the quadratic is of the form <strong>a\u00b2x\u00b2 + 2abx + b\u00b2<\/strong>, it factors as <strong>(ax + b)\u00b2<\/strong>.<\/li>\n\n\n\n<li><strong>Difference of Squares:<\/strong> If the quadratic is <strong>a\u00b2x\u00b2 &#8211; b\u00b2<\/strong>, it factors as <strong>(ax &#8211; b)(ax + b)<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-color has-link-color has-large-font-size wp-elements-d0744e0bac322555d611e54dc8b4a2d6\" style=\"color:#000060\"><strong>Checking Your Answer<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Expand the factorized expression to verify if it equals the original quadratic.<\/li>\n\n\n\n<li>If incorrect, revisit the factor pairs and adjust the grouping.<\/li>\n<\/ul>\n\n\n\n<p class=\"has-large-font-size\">To&nbsp;factorise a quadratic of the form&nbsp;ax2+bx+c,&nbsp;write it&nbsp;as<\/p>\n\n\n\n<p class=\"has-large-font-size\">ax<sup>2<\/sup>+r1x+r2x+c<\/p>\n\n\n\n<p class=\"has-large-font-size\">where&nbsp;a . c=r1 . r2&nbsp;and&nbsp;b=r1+r2.&nbsp;Then factor by&nbsp;grouping.<\/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:#fbe3b9\"><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-bf028e21fa0dc48a114d37bb478c3e7d\" style=\"color:#b00012\">\ud83c\udfc0 Factorise.<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+13f+11=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+13f+11<\/p>\n\n\n\n<p>The&nbsp;product&nbsp;ac&nbsp;is&nbsp;22,&nbsp;so you need to find a pair of factors with a product of&nbsp;22.&nbsp;The&nbsp;b&nbsp;term is&nbsp;13,&nbsp;so you need to find a pair of factors with a sum of&nbsp;13.&nbsp;Since the product is positive&nbsp;(22)&nbsp;and the sum is positive&nbsp;(13),&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;22,&nbsp;and then find the one with a sum of&nbsp;13.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675750420194_862\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;ac=22<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td>1 . 22=22<\/td><td>1+22=23<\/td><\/tr><tr><td><strong>2 . <\/strong><strong>11<\/strong><strong>=<\/strong><strong>22<\/strong><\/td><td><strong>2<\/strong>+<strong>11<\/strong><strong>=<\/strong><strong>13<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;2&nbsp;and&nbsp;11&nbsp;have a sum of&nbsp;13.&nbsp;So, replace the quadratic&#8217;s&nbsp;13f&nbsp;term with&nbsp;2f&nbsp;and&nbsp;11f,&nbsp;and then factor by&nbsp;grouping.<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+13f+11<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+<strong>2<\/strong>f+<strong>11<\/strong>f+11<\/p>\n\n\n\n<p>2f(f+1)+11(f+1)     Factor&nbsp;by grouping; the expressions in brackets should&nbsp;match<\/p>\n\n\n\n<p>(2f+11)(f+1)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(2f+11)(f+1)<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+11f+2f+11       Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>2f<sup>2<\/sup>+13f+11<\/p>\n\n\n\n<p>Yes,&nbsp;2f<sup>2<\/sup>+13f+11=(2f+11)(f+1).<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#d0f2ba\"><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-424ab28637819039234d26b329dcad9d\" style=\"color:#b00012\">\ud83c\udfc0 <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>2m<sup>2<\/sup>+13m+11=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>2m<sup>2<\/sup>+13m+11<\/p>\n\n\n\n<p>The&nbsp;product&nbsp;ac&nbsp;is&nbsp;22,&nbsp;so you need to find a pair of factors with a product of&nbsp;22.&nbsp;The&nbsp;b&nbsp;term is&nbsp;13,&nbsp;so you need to find a pair of factors with a sum of&nbsp;13.&nbsp;Since the product is positive&nbsp;(22)&nbsp;and the sum is positive&nbsp;(13),&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;22,&nbsp;and then find the one with a sum of&nbsp;13.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675758910210_870\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;ac=22<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td>1 . 22=22<\/td><td>1+22=23<\/td><\/tr><tr><td><strong>2 . <\/strong><strong>11<\/strong><strong>=<\/strong><strong>22<\/strong><\/td><td><strong>2<\/strong>+<strong>11<\/strong><strong>=<\/strong><strong>13<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;2&nbsp;and&nbsp;11&nbsp;have a sum of&nbsp;13.&nbsp;So, replace the quadratic&#8217;s&nbsp;13m&nbsp;term with&nbsp;2m&nbsp;and&nbsp;11m,&nbsp;and then factor by&nbsp;grouping.<\/p>\n\n\n\n<p>2m<sup>2<\/sup>+13m+11<\/p>\n\n\n\n<p>2m<sup>2<\/sup>+<strong>2<\/strong>m+<strong>11<\/strong>m+11<\/p>\n\n\n\n<p>2m(m+1)+11(m+1)      Factor&nbsp;by grouping; the expressions in brackets should&nbsp;match<\/p>\n\n\n\n<p>(2m+11)(m+1)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(2m+11)(m+1)<\/p>\n\n\n\n<p>2m<sup>2<\/sup>+11m+2m+11    Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>2m<sup>2<\/sup>+13m+11<\/p>\n\n\n\n<p>Yes,&nbsp;2m<sup>2<\/sup>+13m+11=(2m+11)(m+1).<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#ecedad\"><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-424ab28637819039234d26b329dcad9d\" style=\"color:#b00012\">\ud83c\udfc0 <strong>Factorise.<\/strong><\/p>\n\n\n\n<p>2u<sup>2<\/sup>+9u+10=&#8212;&#8212;<\/p>\n<\/div><\/div>\n\n\n\n<p>Look&nbsp;at the given&nbsp;quadratic:<\/p>\n\n\n\n<p>2u<sup>2<\/sup>+9u+10<\/p>\n\n\n\n<p>The&nbsp;product&nbsp;ac&nbsp;is&nbsp;20,&nbsp;so you need to find a pair of factors with a product of&nbsp;20.&nbsp;The&nbsp;b&nbsp;term is&nbsp;9,&nbsp;so you need to find a pair of factors with a sum of&nbsp;9.&nbsp;Since the product is positive&nbsp;(20)&nbsp;and the sum is positive&nbsp;(9),&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;20,&nbsp;and then find the one with a sum of&nbsp;9.<\/p>\n\n\n\n<figure id=\"yui_3_18_1_1_1675759279059_876\" class=\"wp-block-table\"><table><tbody><tr><td>Factor&nbsp;pairs of&nbsp;ac=20<\/td><td>Sum&nbsp;of factor&nbsp;pairs<\/td><\/tr><tr><td>1 .  20=20<\/td><td>1+20=21<\/td><\/tr><tr><td>2 . 10=20<\/td><td>2+10=12<\/td><\/tr><tr><td><strong>4 . <\/strong><strong>5<\/strong><strong>=<\/strong><strong>20<\/strong><\/td><td><strong>4<\/strong>+<strong>5<\/strong><strong>=<\/strong><strong>9<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>The&nbsp;factors&nbsp;4&nbsp;and&nbsp;5&nbsp;have a sum of&nbsp;9.&nbsp;So, replace the quadratic&#8217;s&nbsp;9u&nbsp;term with&nbsp;4u&nbsp;and&nbsp;5u,&nbsp;and then factor by&nbsp;grouping.<\/p>\n\n\n\n<p>2u<sup>2<\/sup>+9u+10<\/p>\n\n\n\n<p>2u<sup>2<\/sup>+<strong>4<\/strong>u+<strong>5<\/strong>u+10<\/p>\n\n\n\n<p>2u(u+2)+5(u+2)<\/p>\n\n\n\n<p>Factor&nbsp;by grouping; the expressions in brackets should&nbsp;match<\/p>\n\n\n\n<p>(2u+5)(u+2)<\/p>\n\n\n\n<p>Finally,&nbsp;check your&nbsp;work.<\/p>\n\n\n\n<p>(2u+5)(u+2)<\/p>\n\n\n\n<p>2u<sup>2<\/sup>+5u+4u+10<\/p>\n\n\n\n<p>Apply&nbsp;the distributive property&nbsp;(FOIL)<\/p>\n\n\n\n<p>2u<sup>2<\/sup>+9u+10<\/p>\n\n\n\n<p>Yes,&nbsp;2u<sup>2<\/sup>+9u+10=(2u+5)(u+2).<\/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\/85390\/331\/970\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-168.png\" alt=\"\" class=\"wp-image-9033\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-168.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-168-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-168-150x150.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\/520\/849\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-173.png\" alt=\"\" class=\"wp-image-9034\" srcset=\"https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-173.png 500w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-173-300x300.png 300w, https:\/\/8thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-173-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Factorise quadratics with other leading coefficients Key Notes: Understanding Quadratic Equations Factorization Process Example 1: Factorize 2x\u00b2 + 7x + 3 Example 2: Factorize 3x\u00b2 &#8211; 8x &#8211; 3 Special Cases Checking Your Answer To&nbsp;factorise a quadratic of the form&nbsp;ax2+bx+c,&nbsp;write it&nbsp;as ax2+r1x+r2x+c where&nbsp;a . c=r1 . r2&nbsp;and&nbsp;b=r1+r2.&nbsp;Then factor by&nbsp;grouping. Learn with an example \ud83c\udfc0 Factorise.<a class=\"more-link\" href=\"https:\/\/8thclass.deltapublications.in\/index.php\/y-4-factorise-quadratics-with-other-leading-coefficients\/\">Continue reading <span class=\"screen-reader-text\">&#8220;Y.4 Factorise quadratics with other leading coefficients&#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-497","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\/497","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=497"}],"version-history":[{"count":21,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/497\/revisions"}],"predecessor-version":[{"id":19945,"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/497\/revisions\/19945"}],"wp:attachment":[{"href":"https:\/\/8thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=497"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}