[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207749-en":3,"doc-seo-207749-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207749,962084925290,"Caleb Sterling","https://ap-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",4,"Exam","GCSE Further Mathematics Revision Guide - worked examples on calculus, algebra and geometry","GCSE Further Mathematics revision guide providing a structured set of worked examples aligned with Further Mathematics specification topics. Covers key methods across Algebra, including indices with rational exponents, quadratic equations and simultaneous equations, plus solution techniques such as substitution and graphical/approximate roots. Extends to expansions and factorisation, the remainder and factor theorems, and core Geometry skills including line equations, distance, gradients, circle properties, and angle facts for semicircles.","THOMAS WHITHAM SIXTH FORM  \nGCSE Further Mathematics  \nRevision Guide  \nS J Cooper  \nThe book contains a number of worked examples covering the topics needed in the Further Mathematics Specifications. This includes Calculus, Trigonometry, Geometry and the more advanced Algebra.  \nAlgebra  \n􀂉  Indices for all rational exponents  \nam 􀁵 an 􀀠 am􀀎n , am 􀁹 an 􀀠 am􀀐n , 􀀋am 􀀌n 􀀠 amn , a 0 􀀠 1 , a ~~ ~~12 􀀠 a  , a  1n 􀀠 ~~n~~a  , a 􀀐n 􀀠 a1n~~ ~~ , ~~ ~~a~~ ~~n~~ ~~ 􀀠 an ,  \n􀂧 a 􀂷  \n􀂨 ~~ ~~ 􀂸  \n􀂩b 􀂹  \n􀀐n  \n􀂧 b 􀂷 n  \n􀀠 􀂨 ~~ ~~ 􀂸  \n􀂩a 􀂹  \nam n 􀀠 nam  􀀠 􀀋na  􀀌m  \nExample 8~~1~~3 􀀠 ~~3~~8 􀀠 2  Example 9 􀀐 ~~ 1~~2 􀀠 ~~ ~~11~~ ~~ 􀀠  1  􀀠  1  \n9 2 ~~ ~~9 3  \nExample x ~~ 2~~3 􀀠 4 􀂟 􀀋x ~~ 2~~3 􀀌32 􀀠 4~~3~~2 􀂟 x 􀀠 􀀋 4 􀀌3 􀀠 8 Example Simplify 2x􀀐1 􀁵 􀀋3x􀀌2 􀁹 6x3  \n2x􀀐1 􀁵 􀀋3x􀀌2 􀁹 6x3 􀀠 2x 􀁵 9x2 􀁵 61x3~~ ~~ 􀀠 18x6x42~~ ~~ 􀀠 x32~~ ~~  \n􀂉  Quadratic equations ax 2 􀀎 bx 􀀎 c 􀀠 0 􀂉  \nExamples (i) 4x2 􀀠 9 (ii) 4x2 􀀠 9x (iii) 4x2 􀀠 9x 􀀐 2  \n(i)  \n2 9 x 􀀠 4  \n􀀿 x 􀀠 􀁲  \n3 2  \nExample Solve  \n(ii) (iii)  \n4x2 􀀐 9x 􀀠 0 4x2 􀀐 9x 􀀎 2 􀀠 0  \nx(4x 􀀐 9) 􀀠 0 (4x 􀀐 1)(x 􀀐 2) 􀀠 0  \n9   1  \n􀀿 x 􀀠 0 or x 􀀠 􀀿 x 􀀠 2 or x 􀀠  \n4 4  \n(i) 2x2 􀀎 x 􀀐 5 􀀠 0 using the method of CTS (ii) 5x2 􀀐 7x 􀀎1 􀀠 0 using the formula.  \n(i) 2x2 􀀎 x 􀀐 5 􀀠 0  \nx 2  \n2 xx 􀀎 2 􀀎 x 􀀎 􀂧􀂨 1  􀂷􀂸 2  \n2 􀂩4 􀂹  \n􀀋x 􀀎 ~~1~~4~~ ~~ 􀀌2  \n􀀋x 􀀎 ~~1~~4~~ ~~ 􀀌2  \nx 􀀎  \n 1  \n4  \nx  \n􀀠  \n􀀠  \n􀀠  \n􀀠  \n􀀠  \n􀀠  \nx 􀁼 1.35, 􀀐1 .85  \n􀂉  Simultaneous equations  \nExample Solve simultaneously 2x 􀀎 3y 􀀠 8, y 􀀠 x 2 􀀐 x 􀀎 2 Here we substitute for y from the second equation into the first 2x 􀀎 3y 􀀠 8 􀀿 2x 􀀎 3􀀋x 2 􀀐 x 􀀎 2􀀌 􀀠 8 􀀿 3x2 􀀐 x 􀀐 2 􀀠 0  \n􀀿 􀀋3x 􀀎 2􀀌􀀋x 􀀐1􀀌 􀀠 0 􀀿 x 􀀠 􀀐 ~~ 2~~3 , 1  \nwhen x 􀀠 􀀐 ~~ 2~~3 , y 􀀠 ~~4~~9 􀀎 ~~2~~3 􀀎 2 􀀠 ~~2~~9~~8 ~~when x 􀀠 1 , y 􀀠 1􀀐1􀀎 2 􀀠 2  \n} Solutions,  \nThe geometrical interpretation here is that the straight line 2x 􀀎 3y 􀀠 8 and the parabola y 􀀠 x 2 􀀐 x 􀀎 2 intersect at points  \n􀂉  Intersection points of graphs to ‘solve’ equations. There are many equations which can not be solved analytically. Approximate roots to equations can be found graphically if necessary.  \nExample What straight line drawn on the same axes as the graph of y 􀀠 x 3 will give the real root of the equation x 3 􀀎 x 􀀐 3 􀀠 0 ?  \nx 3 􀀎 x 􀀐 3 􀀠 0 􀂟 x 3 􀀠 3 􀀐 x  \n􀀿 draw y 􀀠 3 􀀐 x  \nAs can be seen from the sketch there is 0  only one real root 􀁄.  \nExample  \nObtain the points of intersection of the circle and the  \nline  \n( )  \n( ) ( )  \nUsing ~~ ~~  \nHence points are (   ) ( )  \n􀂉  Expansions and factorisation –extensions Example  \n􀀋2x 􀀐1􀀌􀀋x 2 􀀐 x 􀀎 3􀀌 􀀠 2x3 􀀐 2x2 􀀎 6x  \n􀀐 x 2 􀀎 x 􀀐 3  \n􀀠 2x3 􀀐 3x2 􀀎 7x 􀀐 3  \nExpanding  \nExample x 3 􀀐 9x 􀀠 x􀀋x 2 􀀐 9􀀌 􀀠 x􀀋x 􀀐 3􀀌􀀋x 􀀎 3􀀌 Factorising  \n􀂉 The remainder Theorem  \nIf the polynomial p(x) be divided by 􀀋ax 􀀎 b􀀌 the remainder will be p􀀋􀀐 ~~b~~a~~ ~~ 􀀌  \nExample When p(x) 􀀠 2x3 􀀐 3x 􀀐 5 is divided by 2x 􀀎1 the  \nremainder is p􀀋􀀐 ~~1~~2~~ ~~ 􀀌 􀀠 􀀐 ~~ 1~~4 􀀎 ~~3~~2 􀀐 5 􀀠 􀀐 ~~1~~4~~5 ~~Example Find the remainder when is  \ndivided by .  \nf 􀀋1􀀌 􀀠 4􀀋1􀀌3 􀀐 3􀀋1􀀌2 􀀎11􀀋1􀀌 􀀐 2 􀀠 10 􀀿  remainder is 10  \n􀂉 The factor theorem Following on from the last item p􀀋􀀐 ~~b~~a~~ ~~ 􀀌 􀀠 0 􀂟 ax 􀀎 b is a factor of p(x)  \nExample Show that 􀀋x 􀀐 2􀀌 is a factor of 6x3 􀀐13x2 􀀎 x 􀀎 2 and hence solve the equation 6x3 􀀐13x2 􀀎 x 􀀎 2 􀀠 0  \nLet p(x) 􀀠 6x3 􀀐13x2 􀀎 x 􀀎 2  \np(2) 􀀠 6􀀋2􀀌3 􀀐13􀀋2􀀌2 􀀎 􀀋2􀀌􀀎 2 􀀠 48 􀀐 52 􀀎 2 􀀎 2 􀀠 0  \n􀀿 􀀋x 􀀐 2􀀌 is a factor of p(x)  \np(x) 􀀠 􀀋x 􀀐 2􀀌􀀋6x2 􀀐 x 􀀐1􀀌 …by inspection  \n􀀠 􀀋x 􀀐 2􀀌􀀋3x 􀀎1􀀌􀀋2x 􀀐1􀀌  \n􀀿 Solutions to the equation are x 􀀠 2, 􀀐 ~~1~~3 , ~~1~~2  \nGeometry  \nc 􀁔  \nGradient = m (= tan 􀁔)  \ny  \n􀂉  Gradient/ intercept form of a straight line Equation  \ny 􀀠 mx 􀀎 c  \n0 x  \n􀂉  Distance between two points Given A 􀀋x1 , y1 􀀌 B 􀀋x2 , y2 􀀌 then  \nAB 2 􀀠 􀀋x2 􀀐 x1 􀀌2 􀀎 􀀋y2 􀀐 y1 􀀌2  \n􀂉  Gradient of a line through two points…A􀀋x1 , y1 􀀌 and B􀀋x2 , y2 􀀌 say  \nm 􀀠 y2 􀀐 y1  \nx2 􀀐 x1  \n􀂉  Equation of a line through (x’, y’) of gradient m  y 􀀐 y 􀁣 􀀠 m􀀋x 􀀐 x􀁣 􀀌  \n􀂉  Equation of a line through two points  \nFind the gradient using m 􀀠 y~~ ~~2x~~ ~~2~~ ~~􀀐~~ ~~yx11 and use the formula as above.  \n􀂉  Parallel and ","cbCaigNrQKDK2yR1","https://ap.wps.com/l/cbCaigNrQKDK2yR1","pdf",1340958,1,36,"English","en",105,"# Algebra\n## Indices and rational exponents\n## Quadratic equations\n## Simultaneous equations\n## Graph intersections for approximate solutions\n## Expansions and factorisation\n## Remainder theorem\n## Factor theorem\n# Geometry\n## Straight line equations and gradients\n## Distance, midpoint, parallel and perpendicular lines\n## Circle properties","[{\"question\":\"Which main Further Mathematics topics does the revision guide cover?\",\"answer\":\"It covers Algebra topics such as indices, quadratic and simultaneous equations, expansions and factorisation, and the remainder and factor theorems. It also covers Geometry topics including gradients, line equations, and circle properties.\"},{\"question\":\"How does the guide suggest solving simultaneous equations?\",\"answer\":\"It demonstrates solving simultaneously by substitution—replacing one variable using the second equation and then solving the resulting equation.\"},{\"question\":\"What Geometry concepts are included for lines and circles?\",\"answer\":\"It includes gradient/intercept and general forms of a line, distance between points, midpoint, parallel/perpendicular conditions, and circle facts such as angles in a semicircle and center-radius relationships.\"}]","GCSE Further Mathematics Revision Guide - worked examples on calculus, algebra and geometry | PDF",1788600994,91,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"gcse-further-mathematics-revision-guide-worked-examples-on-calculus-algebra-and-geometry","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/gcse-further-mathematics-revision-guide-worked-examples-on-calculus-algebra-and-geometry/207749/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Which main Further Mathematics topics does the revision guide cover?","Question",{"text":74,"@type":75},"It covers Algebra topics such as indices, quadratic and simultaneous equations, expansions and factorisation, and the remainder and factor theorems. It also covers Geometry topics including gradients, line equations, and circle properties.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the guide suggest solving simultaneous equations?",{"text":79,"@type":75},"It demonstrates solving simultaneously by substitution—replacing one variable using the second equation and then solving the resulting equation.",{"name":81,"@type":72,"acceptedAnswer":82},"What Geometry concepts are included for lines and circles?",{"text":83,"@type":75},"It includes gradient/intercept and general forms of a line, distance between points, midpoint, parallel/perpendicular conditions, and circle facts such as angles in a semicircle and center-radius relationships.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,103,108,113,118,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":101,"slug":102},70,"exam",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":46,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},8,"Research & Report",30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":104,"slug":138},19,"General","general"]