[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571565-105":53,"doc-detail-571565-en":126},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":119,"head_meta":121,"extra_data":123,"updated_unix":125},105,"en","paradigm-a-math-consolidation-cheatsheet-quadratic-equations-inequalities-nature-of-roots-surds-polynomials","Paradigm A Math Consolidation Cheatsheet - Quadratic Equations & Inequalities - Nature of Roots, Surds, Polynomials","","Math consolidation cheatsheet focused on key algebra and calculus-adjacent techniques: completing the square for equations and inequalities, dividing negative coefficients and flipping inequality signs, and solving quadratic inequality ranges. It also summarizes the nature of roots using discriminant conditions and graph behavior. Further sections cover surd rationalisation (eliminating square roots from denominators), mensuration formulas using surds, and core polynomial tools including coefficients, degree, polynomial division with missing terms, identities, remainder and factor theorems, and strategies for solving cubic factorisation and equations.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/paradigm-a-math-consolidation-cheatsheet-quadratic-equations-inequalities-nature-of-roots-surds-polynomials/571565/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/paradigm-a-math-consolidation-cheatsheet-quadratic-equations-inequalities-nature-of-roots-surds-polynomials/571565.png","ImageObject",442,249,{"name":88,"@type":89},"Taylor Morgan","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-23","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":47},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How do you solve quadratic inequalities using this sheet’s method?","Question",{"text":108,"@type":109},"Flush all terms to the left, rearrange into the form of a quadratic using the given coefficients, then solve the quadratic inequalities. Reject cases if required.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"What determines the nature of roots for a quadratic?",{"text":113,"@type":109},"Use the discriminant expression: if it’s less than 0 there are no real roots; equal to 0 gives one repeated root; greater than 0 gives two real roots (distinct). Inequalities can also be interpreted via curve/axis position.",{"name":115,"@type":106,"acceptedAnswer":116},"How is surd rationalisation handled here?",{"text":117,"@type":109},"Rationalise by removing square roots from the denominator. Change signs appropriately while multiplying to eliminate irrational denominators, then simplify the resulting expression.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},571565,1792382376,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":47,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":76},1099523885336,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","EQUATION & INEQUALITIES Completing The Square  \n􀫛􀢞􀫛 − 􀫞􀢞 + 􀫢􀫞   􀫢  \n= 􀫛 (􀢞􀫛 − 􀢞 + )􀫛 􀫛  \n= 􀫛 [(􀢞􀫛 − 􀫞􀫛 􀢞 + ( − 􀫞􀫝)􀫛 − ( − 􀫞􀫝)􀫛 + 􀫢􀫛]= 􀫛 [(􀢞 − 􀫞􀫝)􀫛 + 􀫝􀫚􀫠􀫟]  \n= 􀫛 (􀢞 − 􀫞􀫝)􀫛 + 􀫝􀫠􀫡  \nQuadratic Inequalities [Divide negative, flip sign if it’s −􀢞􀫛 ]  \nQuestion Type 1: Quadratic Inequalities  \nFind the range of values of 􀝔 for which 􀝔(􀝔 − 12) ≤ 3􀝔(1 + 􀝔)  \nStep 1: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 2: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 3: Solve your quadratic inequalities  \nQuestion Type 2: Reverse Quadratic Inequalities  \nFind the values of 􀝌 and 􀝍 for which 􀝔 \u003C −3 or 􀝔 > 2  \nis the solution set of 􀝔 2 + 􀝍 > −􀝌􀝔 .  \nStep 1: 􀝔 \u003C −3 or 􀝔 > 2  \nStep 2: (􀝔 + 3)(􀝔 − 2) > 0 (Reverse and Form Back Original)  \nStep 3: 􀝔 2 + 􀝔 − 6 > 0 (Expand)  \nStep 4: 􀝔 2 − 6 > −􀝔 (Rearrange according to question)  \nStep 5: 􀝍 = −6, 􀝌 − 1 (Compare coefficient)  \nNATURE OF ROOTS  \n\n| Determinants (Curve & Axis) |  |  |\n| --- | --- | --- |\n| 􀜾2 − 4􀜽􀜿 \u003C 0\u003Cbr>􀜾2 − 4􀜽􀜿 = 0\u003Cbr>􀜾2 − 4􀜽􀜿 > 0\u003Cbr>􀜾2 − 4􀜽􀜿 ≥ 0 | No Roots\u003Cbr>1 Roots\u003Cbr>2 Roots\u003Cbr>1/2 Roots | No Real Roots or Imaginary Roots or\u003Cbr>Graph is always positive (Completely above x-axis) or\u003Cbr>Graph is always negative (Completely below x-axis) Real & Equal or\u003Cbr>Real & Repeated Roots Real & Distinct Roots or Different Roots\u003Cbr>Graph has real roots or Graph Intersects the x-axis |\n\nQuestion Type 1: Nature of Roots  \nFind the range of 􀜽, for which the graph is \u003CCondition>  \nStep 1: Equate Curve and Line if applicable* (=, >, \u003C)  \nStep 2: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 3: Determine Nature of Roots base on the condition Step 4: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 5: Solve your quadratic inequalities  \nStep 6: Reject if Necessary  \nQuestion Type 2: Proving Questions (Prove/Show/Explain)  \nShow that the expression is negative for all real values of 􀝔 . Prove that the expression is always positive for all values of 􀝔 .  \nStep 1: Flush everything to the left and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 2: Apply 􀜾2 − 4􀜽􀜿 but don’t put the sign  \nStep 3: Simplify and rearrange according to 􀜽􀝔2 + 􀜾􀝔 + 􀜿  \nStep 4: Use Completing The Square/otherwise (See Condition) Step 5: For all real values of 􀝔 ,   is positive/negative (Proven)  \nExample  \n􀜾 2 − 4􀜽􀜿 = [−(􀝇 + 2)]2 − 4(3)(2􀝇 − 9)  \n= 􀝇 2 + 4􀝇 + 4 − 24􀝇 + 108  \n= 􀝇 2 − 20􀝇 + 111  \n= (􀝇 − 10)2 + 12  \nSince (􀝇 − 10)2 + 12 > 0, 􀜾 2 − 4􀜽􀜿 > 0 and line intersects the curve for all real values ofk.  \nSURDS  \n| Rationalisation of Surds: (We don’t like SqRoots in Denominator)\u003Cbr>Note: Change Signs while Rationalizing |  |\n| --- | --- |\n| \u003Cbr>3 √2 3√2 ~~  ~~ × ~~  ~~ = ~~ ~~√2 √2 2\u003Cbr>\u003Cbr>1 √3 √3 ~~  ~~ × ~~   ~~ = ~~  ~~5√3 √3 15 | \u003Cbr>4 2 − √3 4(2 − √3) ~~  ~~ × ~~  ~~ = ~~ ~~\u003Cbr>2 + √3 2 − √3   4 − 3 = 8 − √3 |\n\n􀜽√􀜾 + 􀜿√􀝀 = 􀝔√􀜾 + 􀝕√􀝀 implies that 􀜽 = 􀝔 and 􀜿 = 􀝕  \nE.g. 3􀜽 + 􀜾 + 2􀜽√5 = −29 − 12√5  \n2􀜽 = −12 ⟹ 􀢇 = −􀫟 3􀜽 + 􀜾 = −29 ⟹ 3(−6) + 􀜾 = −29 ⟹ 􀢈 = −􀫚􀫚  \nMensuration with Surds  \n\n| Cone\u003Cbr>Surface Area: 􀟨􀝎2 + 􀟨􀝎􀝈\u003Cbr>Volume:13 􀟨􀝎2 ℎ | Pyramid Surface Area: Add all sides\u003Cbr>Volume: 13 Base area x Height |\n| --- | --- |\n| Hemi-Sphere Surface Area: 2􀟨􀝎2 [open] /3􀟨􀝎2\u003Cbr>Volume: 23 􀟨􀝎3 | Sphere Surface Area: 4􀟨􀝎2\u003Cbr>Volume: 43 􀟨􀝎3 |\n| Prism Surface Area: Add all sides\u003Cbr>Volume: Cross Section x Length | Cylinder Surface Area: 2􀟨􀝎2 + 2􀟨􀝎ℎ\u003Cbr>Volume: 􀟨􀝎2 ℎ |\n\nPOLYNOMIALS  \nImportant terms:  \nCoefficient of a Variable:  \nNumber in front of the variable (algebra)  \n2 is the coefficient of 2􀝔 , 4 is the coefficient od 4􀝔2  \nDegree of a Polynomial:  \nThe degree of a polynomial is the highest power in the equation  \nAdding, Subtraction & Multiplication follows algebra rules Division of Polynomials  \nNote: When you encounter polynomials such as  \nDivide 􀝔 3 − 9􀝔 + 10 by 􀝔 + 1  \nPlease write it as 􀝔 3 + 0􀝔2 − 9􀝔 + 10. Fil up the missing terms with 0  \nQuestion Type 1: Solving Unknowns  \nMethod 1: Fully expand and compare coefficient  \nMethod 2: Substitution Metho","cbCaifu9Zb1yhO55","https://ap.wps.com/l/cbCaifu9Zb1yhO55","pdf",4667451,8,"English","# Equation & Inequalities\n## Completing The Square\n## Quadratic Inequalities\n## Reverse Quadratic Inequalities\n## Nature of Roots\n## Proving Questions\n# Surds\n## Rationalisation of Surds\n## Mensuration with Surds\n# Polynomials\n## Important Terms\n## Solving Unknowns\n## Remainder & Factor Theorem\n## Reverse Engineer Polynomial Equation\n## Cubic Factorisation\n## Solve Cubic Equations","[{\"question\":\"How do you solve quadratic inequalities using this sheet’s method?\",\"answer\":\"Flush all terms to the left, rearrange into the form of a quadratic using the given coefficients, then solve the quadratic inequalities. Reject cases if required.\"},{\"question\":\"What determines the nature of roots for a quadratic?\",\"answer\":\"Use the discriminant expression: if it’s less than 0 there are no real roots; equal to 0 gives one repeated root; greater than 0 gives two real roots (distinct). Inequalities can also be interpreted via curve/axis position.\"},{\"question\":\"How is surd rationalisation handled here?\",\"answer\":\"Rationalise by removing square roots from the denominator. Change signs appropriately while multiplying to eliminate irrational denominators, then simplify the resulting expression.\"}]","Paradigm A Math Consolidation Cheatsheet - Quadratic Equations & Inequalities - Nature of Roots, Surds, Polynomials | PDF",1791922374]