[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207809-en":3,"doc-seo-207809-105":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},207809,7971474921005,"Grenda","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",4,"Exam","Quadratic Equations - solving methods and practice","Quadratic Equations unit explains what makes an equation quadratic and how to identify its standard form ax2 + bx + c = 0, including the roles of a, b, and c. It then guides learners through four solution techniques: factorisation, completing the square, using the quadratic formula, and solving by drawing graphs. Clear key points emphasize that a cannot be zero and that quadratic equations yield two solutions, supporting mastery through repeated practice exercises.","Quadratic Equations  \nmc-TY-quadeqns-1  \nThis unit is about the solution of quadratic equations. These take the form ax2 +bx+c = 0 . We will look at four methods: solution by factorisation, solution by completing the square, solution using a formula, and solution using graphs  \nIn order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.  \nAfter reading this text, and/or viewing the video tutorial on this topic, you should be able to:  \n• solve quadratic equations by factorisation  \n• solve quadratic equations by completing the square  \n• solve quadratic equations using a formula  \n• solve quadratic equations by drawing graphs  \nContents  \n1. Introduction 2  \n2. Solving quadratic equations by factorisation 2  \n3. Solving quadratic equations by completing the square 5  \n4. Solving quadratic equations using a formula 6  \n5. Solving quadratic equations by using graphs 7  \n[www.mathcentre.ac.uk 1](www.mathcentre.ac.uk 1) 􀀍c mathcentre 2009   \n1. Introduction  \nThis unit is about how to solve quadratic equations. A quadratic equation is one which must contain a term involving x2 , e.g. 3x2 , −5x2 or just x2 on its own. It may also contain terms involving x, e.g. 5x or −7x, or 0.5x. It can also have constant terms-these are just numbers:  \n6 , −7 , 12 .  \nIt cannot have terms involving higher powers of x, like x3 . It cannot have terms like 1x in it. In general a quadratic equation will take the form  \nax2 + bx + c = 0  \na can be any number excluding [zero.](zero. b and)[ b](zero. b and)[ and](zero. b and) c can be any numbers including zero. If b or c is zero then these terms will not appear.  \n Key Point  \nA quadratic equation takes the form  \nax2 + bx + c = 0  \nwhere a , b and c are numbers. The number a cannot be zero.  \nIn this unit we will look at how to solve quadratic equations using four methods:  \n• solution by factorisation  \n• solution by completing the square  \n• solution using a formula  \n• solution using graphs  \nFactorisation and use of the formula are particularly important.  \n2. Solving quadratic equations by factorisation  \nIn this section we will assume that you already know how to factorise a quadratic expression. If this is not the case you can study other material in this series where factorisation is explained.  \nExample  \nSuppose we wish to solve 3x2 = 27 .  \nWe begin by writing this in the standard form of a quadratic equation by subtracting 27 from each side to give 3x2 − 27 = 0 .  \n[www.mathcentre.ac.uk 2](www.mathcentre.ac.uk 2) 􀀍c mathcentre 2009   \nWe now look for common factors. By observation there is a common factor of 3 in both terms. This factor is extracted and written outside a pair of brackets. The contents of the brackets are adjusted accordingly:  \n3x2 − 27 = 3(x2 − 9) = 0  \nNotice here the difference of two squares which can be factorised as  \n3(x2 − 9) = 3(x − 3)(x + 3) = 0  \nIf two quantities are multiplied together and the result is zero then either or both of the quantities must be zero. So either  \nx − 3 = 0 or x + 3 = 0  \nso that  \nx = 3 or x = −3  \nThese are the two solutions of the equation.  \nExample  \nSuppose we wish to solve 5x2 + 3x = 0 .  \nWe look to see if we can spot any common factors. There is a common factor of x in both terms. This is extracted and written in front of a pair of brackets:  \nx(5x + 3) = 0  \nThen either x = 0 or 5x + 3 = 0 from which x = − 35 . These are the two solutions.  \nIn this example there is no constant term. A common error that students make is to cancel the common factor of x in the original equation:  \n5x + 3= 0 so that 5x + 3 = 0 giving x = − ~~ ~~35  \nBut if we do this we lose the solution x = 0 . In general, when solving quadratic equations weare looking for two solutions.  \nExample  \nSuppose we wish to solve x2 − 5x + 6 = 0 .  \nWe factorise the quadratic by looking for two numbers which multiply together to give 6, and add to give −5 . Now  \n−3 × −2 = 6 − 3 + −2 =","cbCaigxHXozLRfv4","https://ap.wps.com/l/cbCaigxHXozLRfv4","pdf",317044,1,9,"English","en",105,"# Introduction\n# Solving quadratic equations by factorisation\n# Solving quadratic equations by completing the square\n# Solving quadratic equations using a formula\n# Solving quadratic equations by using graphs","[{\"question\":\"What is the standard form of a quadratic equation in this unit?\",\"answer\":\"A quadratic equation takes the form ax2 + bx + c = 0. The coefficient a must be non-zero, while b and c can be any numbers (including zero).\"},{\"question\":\"How many solutions does a quadratic equation have?\",\"answer\":\"Quadratic equations are aimed at producing two solutions. Depending on the factors, solutions may be different or repeated.\"},{\"question\":\"What four methods does the unit teach for solving quadratic equations?\",\"answer\":\"The unit covers solving by factorisation, completing the square, using a formula (quadratic formula), and solving using graphs.\"}]","Quadratic Equations - solving methods and practice | PDF",1788601211,23,{"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},"quadratic-equations-solving-methods-and-practice","",{"@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/quadratic-equations-solving-methods-and-practice/207809/",{"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},"What is the standard form of a quadratic equation in this unit?","Question",{"text":74,"@type":75},"A quadratic equation takes the form ax2 + bx + c = 0. The coefficient a must be non-zero, while b and c can be any numbers (including zero).","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How many solutions does a quadratic equation have?",{"text":79,"@type":75},"Quadratic equations are aimed at producing two solutions. Depending on the factors, solutions may be different or repeated.",{"name":81,"@type":72,"acceptedAnswer":82},"What four methods does the unit teach for solving quadratic equations?",{"text":83,"@type":75},"The unit covers solving by factorisation, completing the square, using a formula (quadratic formula), and solving using graphs.","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},[92,96,100,103,108,113,118,123,127,130,134],{"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":21,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":104,"slug":137},19,"General","general"]