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56$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=product}{\\color{@sky}{7\\times(-8)=-56}}"},{"latex":"\\htmlData{anchor=sum}{\\color{@orange}{7+(-8)=-1}}"},{"latex":"a^2-a-56=\\htmlData{anchor=factors}{\\color{@green}{(a+7)(a-8)}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"factors"}]},"order":1,"title":"Write the brackets","description":"Use 7 and -8: their product is -56 and their sum is -1. Writing the factorisation as two brackets earns M1; using 7 and -8 correctly earns A1."}]}],"generatedAt":"2026-07-21T02:50:00.204Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"44725e1e-2bc6-45ae-b7dd-9628c4ccc83e","label":"b","content":"Solve $a^2 - a - 56 = 0$","markscheme":"- Correct solutions: $a = -7$ or $a = 8$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States $a = -7$ and $a = 8$.","code":"B"}],"finalAnswers":["a = -7, a = 8"],"generatedAt":"2026-05-07T01:58:33.588Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"factorising","label":"Factorise the quadratic","steps":[{"type":"text","order":0,"title":"Find the factor pair","content":"Find two numbers whose **product** is $-56$ and whose **sum** is $-1$.\n\nThe pair is $-8$ and $7$: their product is $-56$ and their sum is $-1$. These are the numbers needed to factorise the quadratic.","highlights":[{"text":"$$a^2 - a - 56 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"a^2-a-56=0"},{"latex":"(a\\htmlData{anchor=factor_eight}{-8})(a+7)=0"}],"highlights":[{"color":"sky","style":"box","anchor":"factor_eight"}]},"order":1,"title":"Write the factorisation","description":"Put the two numbers from the factor pair into brackets."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(a\\htmlData{anchor=solve_minus}{-8})(a+7)=0"},{"latex":"a-8=0"},{"latex":"a=8"},{"latex":"a+7=0"},{"latex":"a=-7"}],"highlights":[{"color":"sky","style":"box","anchor":"solve_minus"}]},"order":2,"title":"Solve each factor","description":"If two factors multiply to zero, at least one factor must be zero. Solve each bracket separately."},{"type":"text","order":3,"title":"State both solutions","content":"Therefore,\n\n$$\\boxed{a=-7\\text{ or }a=8}$$\n\nBoth solutions are needed. This earns the single mark.","highlights":[{"text":"Solve $a^2 - a - 56 = 0$","location":"specification"}]}]},{"id":"quadratic_formula","label":"Use the quadratic formula","steps":[{"type":"text","order":0,"title":"Identify the coefficients","content":"Write the quadratic in the form $Aa^2+Ba+C=0$. Comparing with the question gives\n\n$$A=1,\n\\qquad B=-1,\n\\qquad C=-56.$$\n\nBe especially careful that the coefficient of $a$ is negative.","highlights":[{"text":"$$a^2 - a - 56 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"a=\\frac{-\\htmlData{anchor=coefb}{B}\\pm\\sqrt{B^2-4AC}}{2A}"},{"latex":"a=\\frac{-\\htmlData{anchor=subb}{(-1)}\\pm\\sqrt{(-1)^2-4(1)(-56)}}{2(1)}"}],"connections":[{"to":"subb","from":"coefb","color":"sky","label":"substitute $B=-1$"}]},"order":1,"title":"Substitute into the formula","description":"Use $a=\\dfrac{-B\\pm\\sqrt{B^2-4AC}}{2A}$. Substitute the coefficients, keeping brackets around negative values."},{"type":"hyper_latex","block":{"nodes":[{"latex":"a=\\frac{1\\pm\\sqrt{225}}{2}"},{"latex":"a=\\frac{1\\pm\\htmlData{anchor=root}{15}}{2}"},{"latex":"a=\\frac{1+\\htmlData{anchor=plus15}{15}}{2}=8"},{"latex":"a=\\frac{1-\\htmlData{anchor=minus15}{15}}{2}=-7"}],"connections":[{"to":"plus15","from":"root","color":"green","label":"use plus"},{"to":"minus15","from":"root","color":"orange","label":"use minus"}]},"order":2,"title":"Evaluate both signs","description":"First simplify the square root. Then calculate one value using plus and one using minus."},{"type":"text","order":3,"title":"State both solutions","content":"Therefore,\n\n$$\\boxed{a=-7\\text{ or }a=8}$$\n\nBoth values must be given to earn the B1 mark."}]}],"generatedAt":"2026-09-01T09:12:34.068Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"e80b8d85-1743-429d-b625-def6f45e6922:factor_pairs","label":"(a) Find the factor pair","steps":[{"type":"text","order":0,"title":"Choose the factor pair","content":"For $a^2-a-56$, find two integers whose product is $-56$ and whose sum is $-1$.","highlights":[{"text":"Factorise $a^2 - a - 56$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=product}{\\color{@sky}{7\\times(-8)=-56}}"},{"latex":"\\htmlData{anchor=sum}{\\color{@orange}{7+(-8)=-1}}"},{"latex":"a^2-a-56=\\htmlData{anchor=factors}{\\color{@green}{(a+7)(a-8)}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"factors"}]},"order":1,"title":"Write the brackets","description":"Use 7 and -8: their product is -56 and their sum is -1. Writing the factorisation as two brackets earns M1; using 7 and -8 correctly earns A1."}]},{"id":"44725e1e-2bc6-45ae-b7dd-9628c4ccc83e:factorising","label":"(b) Factorise the quadratic","steps":[{"type":"text","order":0,"title":"Find the factor pair","content":"Find two numbers whose **product** is $-56$ and whose **sum** is $-1$.\n\nThe pair is $-8$ and $7$: their product is $-56$ and their sum is $-1$. These are the numbers needed to factorise the quadratic.","highlights":[{"text":"$$a^2 - a - 56 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"a^2-a-56=0"},{"latex":"(a\\htmlData{anchor=factor_eight}{-8})(a+7)=0"}],"highlights":[{"color":"sky","style":"box","anchor":"factor_eight"}]},"order":1,"title":"Write the factorisation","description":"Put the two numbers from the factor pair into brackets."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(a\\htmlData{anchor=solve_minus}{-8})(a+7)=0"},{"latex":"a-8=0"},{"latex":"a=8"},{"latex":"a+7=0"},{"latex":"a=-7"}],"highlights":[{"color":"sky","style":"box","anchor":"solve_minus"}]},"order":2,"title":"Solve each factor","description":"If two factors multiply to zero, at least one factor must be zero. Solve each bracket separately."},{"type":"text","order":3,"title":"State both solutions","content":"Therefore,\n\n$$\\boxed{a=-7\\text{ or }a=8}$$\n\nBoth solutions are needed. This earns the single mark.","highlights":[{"text":"Solve $a^2 - a - 56 = 0$","location":"specification"}]}]},{"id":"44725e1e-2bc6-45ae-b7dd-9628c4ccc83e:quadratic_formula","label":"(b) Use the quadratic formula","steps":[{"type":"text","order":0,"title":"Identify the coefficients","content":"Write the quadratic in the form $Aa^2+Ba+C=0$. Comparing with the question gives\n\n$$A=1,\n\\qquad B=-1,\n\\qquad C=-56.$$\n\nBe especially careful that the coefficient of $a$ is negative.","highlights":[{"text":"$$a^2 - a - 56 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"a=\\frac{-\\htmlData{anchor=coefb}{B}\\pm\\sqrt{B^2-4AC}}{2A}"},{"latex":"a=\\frac{-\\htmlData{anchor=subb}{(-1)}\\pm\\sqrt{(-1)^2-4(1)(-56)}}{2(1)}"}],"connections":[{"to":"subb","from":"coefb","color":"sky","label":"substitute $B=-1$"}]},"order":1,"title":"Substitute into the formula","description":"Use $a=\\dfrac{-B\\pm\\sqrt{B^2-4AC}}{2A}$. Substitute the coefficients, keeping brackets around negative values."},{"type":"hyper_latex","block":{"nodes":[{"latex":"a=\\frac{1\\pm\\sqrt{225}}{2}"},{"latex":"a=\\frac{1\\pm\\htmlData{anchor=root}{15}}{2}"},{"latex":"a=\\frac{1+\\htmlData{anchor=plus15}{15}}{2}=8"},{"latex":"a=\\frac{1-\\htmlData{anchor=minus15}{15}}{2}=-7"}],"connections":[{"to":"plus15","from":"root","color":"green","label":"use plus"},{"to":"minus15","from":"root","color":"orange","label":"use minus"}]},"order":2,"title":"Evaluate both signs","description":"First simplify the square root. Then calculate one value using plus and one using minus."},{"type":"text","order":3,"title":"State both solutions","content":"Therefore,\n\n$$\\boxed{a=-7\\text{ or }a=8}$$\n\nBoth values must be given to earn the B1 mark."}]}],"generatedAt":"2026-09-01T09:12:34.345Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b3935209-66d6-4aa0-8493-f378abbca4d6","revisionId":"fbacd8f9-facd-43c8-9267-88243fb60c1a","questionNumber":"14","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Solving Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"29aed987-5c4d-4f0d-89ab-baf1465bc415","label":"0","content":"Solve $y^2 - 2y - 35 = 0$","markscheme":"- Factorises the quadratic: $(y + 5)(y - 7) = 0$ M2\n- Correct solutions: $y = -5$ or $y = 7$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies factors of -35 that sum to -2 (5 and -7).","code":"M"},{"id":"p2","marks":1,"description":"Writes the quadratic in factorised form $(y+5)(y-7) = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $y = -5$ and $y = 7$.","code":"A"}],"finalAnswers":["y = -5, y = 7"],"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":"Choose the factor pair","content":"Find two numbers that multiply to $-35$ and add to $-2$.\n\nThey are $5$ and $-7$, since $5\\times(-7)=-35$ and $5+(-7)=-2$.","highlights":[{"text":"Solve $y^2 - 2y - 35 = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"y^2-2y-35=\\htmlData{anchor=original}{0}"},{"latex":"\\htmlData{anchor=factorised}{(y+5)(y-7)=0}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"factorised"}],"connections":[{"to":"factorised","from":"original","color":"sky","label":"factorise","style":"solid"}]},"order":1,"title":"Factorise the quadratic","description":"Put the two numbers into brackets with y. Expanding the brackets gives back the original quadratic. Examiner credit: M2 for (y+5)(y-7)=0."},{"type":"text","order":2,"title":"Use the zero-product rule","content":"If two factors multiply to give $0$, at least one factor must be $0$. So solve each bracket separately:\n\n$$y+5=0\\qquad\\text{or}\\qquad y-7=0.$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=plus_five}{\\color{@orange}{y+5}}=0"},{"latex":"y=\\htmlData{anchor=negative_solution}{\\color{@orange}{-5}}"},{"latex":"\\htmlData{anchor=minus_seven}{\\color{@green}{y-7=0}}"},{"latex":"\\htmlData{anchor=positive_solution}{\\color{@green}{y=7}}"}],"highlights":[{"color":"orange","style":"text-color","anchor":"negative_solution"},{"color":"green","style":"text-color","anchor":"positive_solution"}],"connections":[{"to":"negative_solution","from":"plus_five","color":"orange","label":"subtract 5","style":"solid"},{"to":"positive_solution","from":"minus_seven","color":"green","label":"add 7","style":"solid"}]},"order":3,"title":"Solve each factor","description":"Rearrange each one-step equation to find both possible values of y. Examiner credit: A1 for y = -5 or y = 7."}]}],"generatedAt":"2026-07-21T04:11:54.058Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"62eb9144-cf1d-4dde-9d57-dfc5f0eacd47","revisionId":"33df51a6-7736-4e88-9552-6afec0ed7438","questionNumber":"17","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Solving Quadratics","calculatorUse":null,"isCalculation":true,"parts":[{"id":"c159718c-4f12-4de4-ab39-3c866d458686","label":"0","content":"Solve $m^2 + 13m + 40 = 0$","markscheme":"- Factorises the quadratic: $(m + 5)(m + 8) = 0$ M2\n- Correct solutions: $m = -5$ or $m = -8$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies factors of 40 that sum to 13 (5 and 8).","code":"M"},{"id":"p2","marks":1,"description":"Writes the quadratic in factorised form $(m+5)(m+8) = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $m = -5$ and $m = -8$.","code":"A"}],"finalAnswers":["m = -5, m = -8"],"generatedAt":"2026-05-07T01:58:33.588Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths","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":"05ab8093-6ea6-479e-9714-8bbbf2d95eb6","topicSidebar":"$L3a","topicSlug":"grade-5-5-6-solving-quadratics","topicTitle":"Solving Quadratics"}]}],"$L3b","$L3c"]}]

Solving Quadratics

Question 7

Solving Quadratics Questions

Solving Quadratics

Question 7

Solving Quadratics Questions