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The middle term changes sign, so expand the two brackets separately before subtracting.","highlights":[{"text":"Expand and Simplify","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(2+\\sqrt{3})^2=\\htmlData{anchor=plus_expansion}{\\color{@sky}{2^2+2(2)(\\sqrt{3})+(\\sqrt{3})^2}}$"},{"latex":"$$(2+\\sqrt{3})^2=\\htmlData{anchor=plus_result}{\\color{@sky}{7+4\\sqrt{3}}}$"},{"latex":"$$(2-\\sqrt{3})^2=\\color{@orange}{2^2-2(2)(\\sqrt{3})+(\\sqrt{3})^2}$"},{"latex":"$$(2-\\sqrt{3})^2=\\color{@orange}{7-4\\sqrt{3}}$"},{"latex":"$$(2+\\sqrt{3})^2-(2-\\sqrt{3})^2=(7+4\\sqrt{3})-(7-4\\sqrt{3})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"plus_expansion"},{"color":"sky","style":"text-color","anchor":"plus_result"}],"connections":[{"to":"plus_result","from":"plus_expansion","color":"sky","label":"simplify"}]},"order":1,"title":"Expand both squares","description":"Apply the two square identities carefully. Obtaining both simplified expansions earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(7+4\\sqrt{3})-(7-4\\sqrt{3})$"},{"latex":"$$7+4\\sqrt{3}-7+4\\sqrt{3}$"}]},"order":2,"title":"Remove the bracket","description":"The subtraction sign changes both signs in the second bracket: positive seven becomes negative seven, while negative four root three becomes positive four root three."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$7+4\\sqrt{3}-7+4\\sqrt{3}$"},{"latex":"$$4\\sqrt{3}+4\\sqrt{3}$"},{"latex":"$$8\\sqrt{3}$"}]},"order":3,"title":"Collect like terms","description":"First, seven minus seven cancels. Then combine the two identical surd terms to get eight root three. This correct simplified result earns A1."}]}],"generatedAt":"2026-07-20T16:40:35.231Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":59},"walkthroughVideo":null},{"id":"58da9761-01de-4848-8c89-1b88aeabaae6","revisionId":"c0f9fe60-d31e-4889-94fa-108612bea293","questionNumber":"10","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":false,"parts":[{"id":"3da5f6cb-b6c2-4ac8-af8f-96a0608e7088","label":"0","content":"Rationalise the denominator $\\frac{6}{\\sqrt{3}}$","markscheme":"- Award method mark for multiplying numerator and denominator by $\\sqrt{3}$: $\\frac{6\\sqrt{3}}{3}$ M1\n- Final answer $2\\sqrt{3}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies numerator and denominator by $\\sqrt{3}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of $2\\sqrt{3}$.","code":"A"}],"finalAnswers":["2\\sqrt{3}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":76},"walkthroughVideo":null},{"id":"c00c601a-faa1-4fe7-90a3-343c6368b2cb","revisionId":"3ca8ade5-2320-45ea-bb49-9013c14f3e6e","questionNumber":"11","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Surds","calculatorUse":null,"isCalculation":false,"parts":[{"id":"ed20a8ac-3503-452d-a329-c8fe670cef04","label":"0","content":"Rationalise the denominator $\\frac{x}{\\sqrt{x}}$","markscheme":"- Award method mark for multiplying by $\\frac{\\sqrt{x}}{\\sqrt{x}}$ to get $\\frac{x\\sqrt{x}}{x}$ M1\n- Final answer $\\sqrt{x}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies numerator and denominator by $\\sqrt{x}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer of $\\sqrt{x}$.","code":"A"}],"finalAnswers":["\\sqrt{x}"],"generatedAt":"2026-05-07T02:09:58.160Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"multiply_by_root","label":"Multiply by the root","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{x}{\\sqrt{x}}\\times \\htmlData{anchor=multiplier}{\\color{@sky}{\\dfrac{\\sqrt{x}}{\\sqrt{x}}}}$"},{"latex":"$$\\dfrac{x\\sqrt{x}}{\\htmlData{anchor=root_product}{\\color{@amber}{\\sqrt{x}\\times\\sqrt{x}}}}$"},{"latex":"$$\\dfrac{x\\sqrt{x}}{\\htmlData{anchor=denominator_x}{\\color{@amber}{x}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"multiplier"},{"color":"amber","style":"text-color","anchor":"root_product"},{"color":"amber","style":"text-color","anchor":"denominator_x"}],"connections":[{"to":"denominator_x","from":"root_product","color":"amber","label":"root times root = x"}]},"order":1,"title":"Multiply by the root","description":"Multiply the fraction by $\\dfrac{\\sqrt{x}}{\\sqrt{x}}$. This equals $1$, so the value is unchanged; equivalently, the numerator and denominator are each multiplied by the non-zero factor $\\sqrt{x}$. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{\\htmlData{anchor=num_x}{\\color{@orange}{x}}\\sqrt{x}}{\\htmlData{anchor=den_x}{\\color{@orange}{x}}}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=cancelled_num_x}{\\color{@orange}{\\cancel{x}}}\\sqrt{x}}{\\htmlData{anchor=cancelled_den_x}{\\color{@orange}{\\cancel{x}}}}$"},{"latex":"$$\\boxed{\\htmlData{anchor=answer}{\\color{@green}{\\sqrt{x}}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"num_x"},{"color":"orange","style":"text-color","anchor":"den_x"},{"color":"orange","style":"text-color","anchor":"cancelled_num_x"},{"color":"orange","style":"text-color","anchor":"cancelled_den_x"}]},"order":2,"title":"Cancel the common factor","description":"Cancel the common factor $x$ in the numerator and denominator. The remaining factor is the rationalised answer, earning the accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T16:37:53.342Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":60},"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"c484ee1f-1bf6-47f3-8890-f0e0c9be7909","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"0dacce83-74f3-4f99-8117-8330978282ef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"72342b57-0e03-4584-aeb4-c241090951c7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"cb761f3d-d598-47d0-8d14-da3783309823","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"8e84081a-aed1-439b-8908-c1900fffcb4a","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"8f495886-5503-40d6-8287-a440cec45cef","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"fd3b66f0-2b80-4928-9527-ffa5c326123b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"96b5c553-923a-4ae0-a9cb-0e38ba01685d","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"459ea282-538c-47a6-851e-37e43fa737e7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"58da9761-01de-4848-8c89-1b88aeabaae6","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"c00c601a-faa1-4fe7-90a3-343c6368b2cb","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"a79070ee-d294-4a3d-a2d3-fbcad7907073","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"45b051e4-21f9-4ca0-8299-deb8d6b8daf7","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8589d66b-7925-4d40-8356-acde4f97efda","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"33397bbc-7bcc-4813-8a4b-b51313185314","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a2be29b3-42b1-4a7b-b349-cded36ca8985","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"18eb1e0e-a1ea-4b6b-8395-5fe2b0f1da54","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1f743cac-719e-4a41-b668-629d4a63f8e0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3ef7bcea-af90-4560-9d02-414c974f9ba5","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"e56f66f9-08d3-4347-bc34-22336ea478f4","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true}],"title":"Surds","topicId":"341c0e44-a1d2-46ae-b271-971e5e815956","topicSidebar":"$L3a","topicSlug":"grade-7-7-1-surds","topicTitle":"Surds"}]}],"$L3b","$L3c"]}]