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factors of 108 that sum to -21.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $(2x - 9)(x - 6)$.","code":"A"}],"finalAnswers":["(2x - 9)(x - 6)"],"generatedAt":"2026-05-07T02:06:38.085Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"split_middle_term","label":"Split the middle term","steps":[{"type":"text","order":0,"title":"Find the factor pair","content":"For $2x^2-21x+54$, multiply the coefficient of $x^2$ by the constant: $2\\times54=108$.\n\nFind two factors of $108$ which add to $-21$. They are $-9$ and $-12$, because $(-9)\\times(-12)=108$ and $-9+(-12)=-21$.\n\nThis identifies the required pair and earns the **M1** mark.","highlights":[{"text":"Factorise $2x^2 - 21x + 54$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{\\color{@sky}{2x^2-21x+54}}"},{"latex":"\\htmlData{anchor=split}{\\color{@sky}{2x^2-9x-12x+54}}"},{"latex":"\\htmlData{anchor=grouped}{\\color{@amber}{x(2x-9)-6(2x-9)}}"},{"latex":"\\htmlData{anchor=answer}{\\color{@green}{(2x-9)(x-6)}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"split","from":"original","color":"sky","label":"split using -9, -12","style":"dashed"},{"to":"grouped","from":"split","color":"amber","label":"group terms","style":"dashed"},{"to":"answer","from":"grouped","color":"green","label":"factor common bracket","style":"dashed"}]},"order":1,"title":"Factor by grouping","description":"Split the middle term using the factor pair, then take out the common bracket. The final line is the factorisation, earning A1."}]}],"generatedAt":"2026-07-21T04:19:33.638Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d5f39c2a-88a0-43c6-978c-57d476c1ba4f","revisionId":"f4670ef6-d7d5-465d-9f4d-e14d7b1ce656","questionNumber":"18","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Factorising Harder Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"6fcc50e3-d449-4f3f-a59a-4e704b638bd5","label":"a","content":"Solve $5x^2 - 80 = 0$","markscheme":"- Factorises the expression into $5(x + 4)(x - 4)$ or rearranges to $x^2 = 16$ M1\n- Identifies the need for both positive and negative roots M1\n- Final answers $x = -4$ and $x = 4$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Factorises the expression or rearranges to $x^2 = 16$.","code":"M"},{"id":"p2","marks":1,"description":"Shows intent to find both roots.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $x = -4$ and $x = 4$.","code":"A"}],"finalAnswers":["x = -4, x = 4"],"generatedAt":"2026-05-07T02:06:38.085Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"d7d8fdb6-7e25-4074-b845-414983bf3111","revisionId":"e24fdde8-5816-4448-a6ec-13e24b02124d","questionNumber":"19","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Factorising Harder Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"3a7f5343-20b8-4a34-8c84-251b0654b628","label":"a","content":"Factorise $x^2 + 2xy + y^2$","markscheme":"- Recognises the perfect square trinomial pattern M1\n- Final answer $(x + y)(x + y)$ or $(x + y)^2$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Recognises the structure of a perfect square trinomial.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $(x + y)(x + y)$ or $(x + y)^2$.","code":"A"}],"finalAnswers":["(x + y)^2"],"generatedAt":"2026-05-07T02:06:38.085Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"perfect_square","label":"Recognise a perfect square","steps":[{"type":"text","order":0,"title":"Spot the square pattern","content":"The first and last terms are squares: $x^2=(x)^2$ and $y^2=(y)^2$. The middle term is $2xy$, which is twice the product of $x$ and $y$.\n\nSo this matches the perfect-square pattern $a^2+2ab+b^2=(a+b)^2$. Recognising this pattern earns the **M1**.","highlights":[{"text":"Factorise $x^2 + 2xy + y^2$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=expression}{\\color{@sky}{x^2+2xy+y^2}}"},{"latex":"\\htmlData{anchor=factorised}{\\color{@green}{(x+y)^2}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"expression"},{"color":"green","style":"text-color-with-background","anchor":"factorised"}],"connections":[{"to":"factorised","from":"expression","color":"green","label":"perfect-square pattern","style":"solid"}]},"order":1,"title":"Write the factorisation","description":"The final factorisation shown below earns A1."}]}],"generatedAt":"2026-07-21T04:13:23.268Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"3bb21d01-9a45-4be2-a065-a4a9579601f5","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ee9f9894-2d44-4dd3-ae6b-057f2e161fee","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"4dc9e8d0-c31f-4c54-9ac0-0e607bffb5bd","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0b0feb0e-aae1-41a6-b12f-9caf0922c76e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"a51a9195-19cc-4ffd-8f21-4b478da954e7","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"9ca38bdc-aae3-4446-afff-bca6e8fbff63","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"39e0a108-d7d2-4bcd-b0ee-46742e62458f","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"65e58598-1de1-4e0c-a13b-07edefa8c4f8","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6bf8e53a-f5aa-4e47-bc80-b75668c474a3","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b221100c-3a06-45f6-9e9a-c56bd50f6b6c","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"644f82b0-5ae3-4077-934a-fc2d73b02100","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"eb6956bd-2510-4959-a5ce-c557de9458b0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"23334b17-ffaa-487c-be33-f6d1c6a211c3","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"4157f0a2-a26d-41c7-879c-358678be4cbe","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"19feebda-9403-457c-8fa3-e0792636902b","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c46305ea-f88d-487f-a44e-026584333267","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"f65fd5fc-d81c-4081-a46f-d0a2b9bc59a7","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d5f39c2a-88a0-43c6-978c-57d476c1ba4f","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d7d8fdb6-7e25-4074-b845-414983bf3111","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"7a0f3bd3-b52d-427e-8430-042e02488a5b","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"154ea538-be4c-45df-bd45-0eecd7491ea1","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true}],"title":"Factorising Harder Quadratics","topicId":"33714bef-2511-43cd-8ea5-f98f12dd98fe","topicSidebar":"$L3c","topicSlug":"grade-7-7-5-factorising-harder-quadratics","topicTitle":"Factorising Harder Quadratics"}]}],"$L3d","$L3e"]}]

Factorising Harder Quadratics

Question 5

Factorising Harder Quadratics Questions

Factorising Harder Quadratics

Question 5

Factorising Harder Quadratics Questions