uadratics","calculatorUse":null,"parts":[{"id":"59d757c4-211c-489a-b754-99f4044fa559","label":"a","content":"Solve $15x^2 - 22x + 8 = 0$","markscheme":"- Factorises the quadratic into $(3x - 2)(5x - 4)$ M1\n- Sets factors to zero to find solutions M1\n- Final answers $x = \\frac{2}{3}$ and $x = \\frac{4}{5}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly factorises the quadratic into $(3x - 2)(5x - 4)$.","code":"M"},{"id":"p2","marks":1,"description":"Sets factors to zero to solve for $x$.","code":"M"},{"id":"p3","marks":1,"description":"Final answers $x = \\frac{2}{3}$ and $x = \\frac{4}{5}$.","code":"A"}],"finalAnswers":["x = 2/3, x = 4/5"],"generatedAt":"2026-05-07T02:06:38.085Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"factorising","label":"Factorise then solve","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{15x^2-22x+8=0}$"},{"latex":"$$\\htmlData{anchor=factorised}{(3x-2)(5x-4)=0}$"}],"connections":[{"to":"factorised","from":"original","color":"sky","label":"factorise"}]},"order":1,"title":"Factorise the quadratic","description":"Use 3x and 5x because their product gives 15x². Use -2 and -4 because their product gives +8; the cross-products are -12x and -10x, which total -22x. This correct factorisation earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=product}{(3x-2)(5x-4)=0}$"},{"latex":"$$\\htmlData{anchor=case_left}{3x-2=0}$"},{"latex":"$$\\htmlData{anchor=case_right}{5x-4=0}$"}],"connections":[{"to":"case_left","from":"product","color":"orange","label":"first factor"},{"to":"case_right","from":"product","color":"green","label":"second factor"}]},"order":2,"title":"Set each factor zero","description":"For a product to equal zero, at least one factor must equal zero, so solve both cases. This earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=eq1_start}{3x-2=0}$"},{"latex":"$$\\htmlData{anchor=eq1_add}{3x=2}$"},{"latex":"$$\\htmlData{anchor=eq1_answer}{x=\\frac{2}{3}}$"},{"latex":"$$\\htmlData{anchor=eq2_start}{5x-4=0}$"},{"latex":"$$\\htmlData{anchor=eq2_add}{5x=4}$"},{"latex":"$$\\htmlData{anchor=eq2_answer}{x=\\frac{4}{5}}$"}],"connections":[{"to":"eq1_add","from":"eq1_start","color":"sky","label":"add 2"},{"to":"eq1_answer","from":"eq1_add","color":"sky","label":"divide by 3"},{"to":"eq2_add","from":"eq2_start","color":"green","label":"add 4"},{"to":"eq2_answer","from":"eq2_add","color":"green","label":"divide by 5"}]},"order":3,"title":"Solve both equations","description":"Add 2 to both sides in the first equation, and add 4 to both sides in the second equation. Then divide by the coefficient of x. Give both values for the final answer mark."}]}],"generatedAt":"2026-07-21T04:20:02.020Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"c46305ea-f88d-487f-a44e-026584333267","revisionId":"3978592c-de07-4b07-b680-246b8233adaf","questionNumber":"16","questionType":"LA","specification":"","difficulty":1,"difficultyBand":null,"totalMarks":2,"sourceTitle":"Factorising Harder Quadratics","calculatorUse":null,"parts":[{"id":"8b14bea7-c8c9-456e-965b-9e956536cd20","label":"a","content":"Factorise fully $2x^2 - 98$","markscheme":"- Factors out the common factor 2 to get $2(x^2 - 49)$ M1\n- Uses difference of two squares to get $2(x + 7)(x - 7)$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Factors out the common factor 2.","code":"M"},{"id":"p2","marks":1,"description":"Correctly applies difference of two squares to factorise fully.","code":"A"}],"finalAnswers":["2(x + 7)(x - 7)"],"generatedAt":"2026-05-07T02:06:38.085Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"common_factor_then_difference_squares","label":"Common factor then di

Factorising Harder Quadratics Questions

Factorising Harder Quadratics Questions

Factorising Harder Quadratics Questions

Factorising Harder Quadratics Questions