alMarks":3,"sourceTitle":"Completing the Square","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d776cbdf-8acf-4bfc-9c54-08d07435387c","label":"0","content":"By completing the square, find the coordinates of the turning point of the curve with the equation $y = x^2 + 8x + 3$\nYou must show all your working.","markscheme":"- Completes the square: $(x + 4)^2 - 16 + 3$ M1\n- Simplifies to $(x + 4)^2 - 13$ M1\n- Turning point: $(-4, -13)$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly identifies $(x + 4)^2$ and subtracts $16$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the constant term to $-13$.","code":"M"},{"id":"p3","marks":1,"description":"States the turning point as $(-4, -13)$.","code":"A"}],"finalAnswers":["(-4, -13)"],"generatedAt":"2026-05-07T02:10:18.347Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"completing_the_square","label":"Complete the square","steps":[{"type":"text","order":0,"title":"Halve the coefficient","content":"To complete the square, halve the coefficient of $x$. Half of $8$ is $4$, so the bracket begins $(x+4)^2$.","highlights":[{"text":"By completing the square","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=\\color{@sky}{\\htmlData{anchor=original}{x^2+8x}}+3$"},{"latex":"$$y=\\color{@sky}{\\htmlData{anchor=square}{(x+4)^2}}\\color{@amber}{\\htmlData{anchor=correction}{-16}}+3$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original"},{"color":"sky","style":"text-color","anchor":"square"},{"color":"amber","style":"text-color","anchor":"correction"}]},"order":1,"title":"Balance the expression","description":"Half of 8 is 4. Expanding (x + 4)² gives x² + 8x + 16, so subtract 16 to keep the expression unchanged. This line earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=(x+4)^2+\\color{@orange}{\\htmlData{anchor=constants}{(-16+3)}}$"},{"latex":"$$y=(x+4)^2\\color{@orange}{\\htmlData{anchor=simplified}{-13}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"constants"},{"color":"orange","style":"text-color","anchor":"simplified"}]},"order":2,"title":"Simplify the constant","description":"Combine the constant terms. Reaching the simplified completed-square form earns the second M1 mark."},{"type":"text","order":3,"title":"Read the turning point","content":"The square $(x+4)^2$ has its minimum value $0$ when $x=-4$. Then $y=-13$, so the turning point is\n\n$$\\boxed{(-4,-13)}$$\n\nGiving the coordinates in this order earns the A1 mark.","highlights":[{"text":"find the coordinates of the turning point","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:31:42.186Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":55},"walkthroughVideo":null},{"id":"bd55144c-c5fa-4afd-a063-a4f3d79bf0d9","revisionId":"b639d0fc-2f34-4b21-9d1b-f5d257c0da5f","questionNumber":"6","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Completing the Square","calculatorUse":null,"isCalculation":true,"parts":[{"id":"be9259bd-a7a2-4d68-be14-672ba68be734","label":"0","content":"By completing the square, find the coordinates of the turning point of the curve with the equation $y = x^2 + 10x - 8$\nYou must show all your working.","markscheme":"- Completes the square: $(x + 5)^2 - 25 - 8$ M1\n- Simplifies to $(x + 5)^2 - 33$ M1\n- Turning point: $(-5, -33)$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly identifies $(x + 5)^2$ and subtracts $25$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the constant term to $-33$.","code":"M"},{"id":"p3","marks":1,"description":"States the turning point as $(-5, -33)$.","code":"A"}],"finalAnswers":["(-5, -33)"],"generatedAt":"2026-05-07T0

Completing the Square

Question 3

Completing the Square Questions

Completing the Square

Question 3

Completing the Square Questions