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bg-background","children":["$","$L3a",null,{"backHref":"/igcse/maths/oxford-aqa/questions","basePath":"/igcse/maths/oxford-aqa/grade-5-5-6-solving-quadratics/questions","formulaUrl":"https://www.oxfordaqa.com/oaqaresources/maths/assessment/9260-oxfordaqa-international-gcse-mathematics-formulae-sheet-ext-e3.pdf","initialIndex":11,"initialQuestionWindow":[{"id":"2ada5dd1-ffd8-4c4a-a7cc-c334dd668bd1","revisionId":"2f67cbf7-898a-45be-9863-a7bd02d324da","questionNumber":"11","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Solving Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"2add1e26-2ea0-4019-8beb-c40504dd8170","label":"0","content":"Solve $x^2 + 19x + 90 = 0$","markscheme":"- Factorises the quadratic: $(x + 9)(x + 10) = 0$ M2\n- Correct solutions: $x = -9$ or $x = -10$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies factors of 90 that sum to 19 (9 and 10).","code":"M"},{"id":"p2","marks":1,"description":"Writes the quadratic in factorised form $(x+9)(x+10) = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $x = -9$ and $x = -10$.","code":"A"}],"finalAnswers":["x = -9, x = -10"],"generatedAt":"2026-05-07T01:58:33.588Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factorising","label":"Factorising","steps":[{"type":"text","order":0,"title":"Find the factor pair","content":"To factorise, find two positive numbers that **multiply to** $90$ and **add to** $19$. They are $9$ and $10$.\n\nThe plus signs are needed because both terms in the quadratic are positive.","highlights":[{"text":"Solve $x^2 + 19x + 90 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{x^2+19x+90=0}"},{"latex":"\\htmlData{anchor=factorised}{(x+9)(x+10)=0}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original"},{"color":"sky","style":"text-color-with-background","anchor":"factorised"}],"connections":[{"to":"factorised","from":"original","color":"sky","label":"factorise"}]},"order":1,"title":"Write the factors","description":"Put the two numbers into brackets with $x$. This factorisation earns the M2 mark."},{"type":"text","order":2,"title":"Use the zero product","content":"If two factors multiply to $0$, at least one factor must be $0$. So set each bracket equal to $0$ in turn."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first_bracket}{x+9=0}"},{"latex":"\\htmlData{anchor=first_solution}{x=-9}"},{"latex":"\\htmlData{anchor=second_bracket}{x+10=0}"},{"latex":"\\htmlData{anchor=second_solution}{x=-10}"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"first_solution"},{"color":"green","style":"text-color-with-background","anchor":"second_solution"}],"connections":[{"to":"first_solution","from":"first_bracket","color":"orange","label":"subtract 9"},{"to":"second_solution","from":"second_bracket","color":"green","label":"subtract 10"}]},"order":3,"title":"Solve each bracket","description":"Subtract the number in each bracket to find both possible values of $x$. These correct solutions earn the A1 mark."}]}],"generatedAt":"2026-07-21T04:09:04.708Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"5f32ba50-332a-42a7-996d-1965f7e48ce8","revisionId":"96d6c54c-c782-463a-b41f-a512d06cde92","questionNumber":"12","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Solving Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"37eb367c-0702-4203-b173-7fa45d70ee47","label":"0","content":"Solve $x^2 + 11x - 42 = 0$","markscheme":"- Factorises the quadratic: $(x - 3)(x + 14) = 0$ M2\n- Correct solutions: $x = 3$ or $x = -14$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies factors of -42 that sum to 11 (-3 and 14).","code":"M"},{"id":"p2","marks":1,"description":"Writes the quadratic in factorised form $(x-3)(x+14) = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $x = 3$ and $x = -14$.","code":"A"}],"finalAnswers":["x = 3, x = -14"],"generatedAt":"2026-05-07T01:58:33.588Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factorising","label":"Factorising","steps":[{"type":"text","order":0,"title":"Find the pair","content":"We need two numbers whose **product** is $-42$ and whose **sum** is $11$. They are $-3$ and $14$: $(-3)(14)=-42$ and $-3+14=11$.","highlights":[{"text":"Solve $x^2 + 11x - 42 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{x^2+11x-42=0}"},{"latex":"\\htmlData{anchor=factorised}{(x-3)(x+14)=0}"}],"connections":[{"to":"factorised","from":"original","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Write the factors","description":"Use those two numbers as the constants in two brackets. Reaching the correct factorisation earns the M2 marks."},{"type":"text","order":2,"title":"Apply zero-product rule","content":"If two factors multiply to give $0$, at least one factor must be $0$. Therefore set each bracket equal to $0$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first_start}{\\color{@green}{x-3=0}}"},{"latex":"\\htmlData{anchor=first_end}{\\color{@green}{x=3}}"},{"latex":"\\htmlData{anchor=second_start}{\\color{@orange}{x+14=0}}"},{"latex":"\\htmlData{anchor=second_end}{\\color{@orange}{x=-14}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"first_end"},{"color":"orange","style":"text-color-with-background","anchor":"second_end"}],"connections":[{"to":"first_end","from":"first_start","color":"green","label":"add 3","style":"solid"},{"to":"second_end","from":"second_start","color":"orange","label":"subtract 14","style":"solid"}]},"order":3,"title":"Solve each bracket","description":"Solve the two one-step equations separately. Giving both roots correctly earns the A1 mark."}]}],"generatedAt":"2026-07-21T04:13:28.586Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"30878799-3a3a-4c2d-984d-545b84d7778b","revisionId":"e305cd60-9e36-474d-b678-2b4b8067fa97","questionNumber":"13","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Solving Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"54fde6fc-3fa3-45c8-8597-c0c1f8f30ead","label":"0","content":"Solve $a^2 - 10a + 16 = 0$","markscheme":"- Factorises the quadratic: $(a - 2)(a - 8) = 0$ M2\n- Correct solutions: $a = 2$ or $a = 8$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies factors of 16 that sum to -10 (-2 and -8).","code":"M"},{"id":"p2","marks":1,"description":"Writes the quadratic in factorised form $(a-2)(a-8) = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $a = 2$ and $a = 8$.","code":"A"}],"finalAnswers":["a = 2, a = 8"],"generatedAt":"2026-05-07T01:58:33.588Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factorising","label":"Factorising","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{a^2-10a+16=0}"},{"latex":"\\htmlData{anchor=factorised}{(a-2)(a-8)=0}"}],"connections":[{"to":"factorised","from":"original","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Factorise the quadratic","description":"The pair $2$ and $8$ multiplies to $16$ and adds to $10$, so use negative signs in both brackets. This factorisation earns the M2 mark."},{"type":"text","order":2,"title":"Use the zero product","content":"If two factors multiply to give $0$, at least one factor must equal $0$. So solve each bracket separately.","highlights":[{"text":"Solve $a^2 - 10a + 16 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=firstfactor}{a-2=0}"},{"latex":"\\htmlData{anchor=firstroot}{a=2}"},{"latex":"\\htmlData{anchor=secondfactor}{a-8=0}"},{"latex":"\\htmlData{anchor=secondroot}{a=8}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"firstroot"},{"color":"orange","style":"text-color-with-background","anchor":"secondroot"}],"connections":[{"to":"firstroot","from":"firstfactor","color":"green","label":"add 2"},{"to":"secondroot","from":"secondfactor","color":"orange","label":"add 8"}]},"order":3,"title":"Solve each factor","description":"Set each factor equal to zero. The two resulting values are the solutions, earning the A1 mark."}]}],"generatedAt":"2026-07-21T03:17:42.668Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"8c4a3814-07f4-423e-961b-9108b4e4f481","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2fe541d6-a67a-44eb-9ce4-d6b00ec581ec","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6aeabf1c-53f7-46c3-8e3a-21c54867b5a5","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d076615f-be28-4fd1-b260-315b540ba51b","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"aa9ac8b7-21c2-49c0-96bf-6a92f17abe01","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d296e581-2509-4b68-8c33-935b5c6c36ac","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1c0b7c87-d951-4125-819e-b7cb826593be","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c71b8017-c256-4d93-ad27-580a5b58d5b9","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"7face5c3-4ccc-4d2e-ae9f-d3e8e9df86ce","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b2cf389a-05e6-4dcf-ad80-4867651ec01c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2ada5dd1-ffd8-4c4a-a7cc-c334dd668bd1","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5f32ba50-332a-42a7-996d-1965f7e48ce8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"30878799-3a3a-4c2d-984d-545b84d7778b","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b3935209-66d6-4aa0-8493-f378abbca4d6","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d600099b-0ecf-46f9-a730-89d28e2f26b4","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"df27fdcc-0236-4e33-b25f-ed879ca090fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"62eb9144-cf1d-4dde-9d57-dfc5f0eacd47","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"7ffc1c73-47e1-4b79-a48f-69185e7cde87","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Solving Quadratics","topicId":"0fc9a4da-ed2b-45f7-ae31-aa3798b15b98","topicSidebar":"$L3b","topicSlug":"grade-5-5-6-solving-quadratics","topicTitle":"Solving Quadratics"}]}],"$L3c","$L3d"]}]

Solving Quadratics

Question 6

Solving Quadratics Questions

Solving Quadratics

Question 6

Solving Quadratics Questions