ll,"firstAttemptFullMarks":{"percent":58}},{"id":"9a9de4f4-eeb3-49c5-938d-c02a74b8964a","revisionId":"6f35cb6a-3fec-4f9e-93a2-f93bf49d0562","questionNumber":"13","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"Question Set 2: Quadratics","calculatorUse":null,"parts":[{"id":"e507a9f0-2fd3-44c3-bcd1-72bd96b09cda","label":"0","content":"$$f(x) = 2x^2 + 8x + 1$\n\nFind the values of the constants $a$, $b$ and $c$ such that\n\n$$f(x) = a(x + b)^2 + c$$","markscheme":"- Factor out 2: $2(x^2 + 4x) + 1$. M1\n- Complete the square inside: $2[(x+2)^2 - 4] + 1$. M1\n- Expand: $2(x+2)^2 - 8 + 1 = 2(x+2)^2 - 7$. A1\n- $a = 2, b = 2, c = -7$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Factors out the coefficient of $x^2$.","code":"M"},{"id":"p2","marks":1,"description":"Completes the square for the $x$ terms.","code":"M"},{"id":"p3","marks":1,"description":"Correctly simplifies to find $c$.","code":"A"},{"id":"p4","marks":1,"description":"States values for $a, b, c$.","code":"A"}],"finalAnswers":["a=2, b=2, c=-7"],"generatedAt":"2026-05-12T02:14:06.561Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"complete_square","label":"Complete the square","steps":[{"type":"text","order":0,"title":"Factor out 2","content":"Group the terms containing $x$, then factor out $2$ from them:\n\n$$f(x)=2(x^2+4x)+1.$$\n\nThis earns the first method mark (M1).","highlights":[{"text":"$$f(x) = 2x^2 + 8x + 1$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"x^2+4x=x^2+4x+4-4"},{"latex":"x^2+4x=(x+2)^2-4"},{"latex":"f(x)=2\\left[(x+2)^2-4\\right]+1"}]},"order":1,"title":"Complete the square","description":"Half of 4 is 2. Adding and subtracting 4 makes a perfect square without changing the expression. Rewriting the bracket as 2[(x + 2)² − 4] + 1 earns the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"f(x)=2(x+2)^2+2(-4)+1"},{"latex":"f(x)=2(x+2)^2-8+1"},{"latex":"f(x)=2(x+2)^2-7"}]},"order":2,"title":"Simplify the constants","description":"Distribute the 2 to the negative 4, then combine the two constants. This earns the accuracy mark (A1)."},{"type":"text","order":3,"title":"Read off the constants","content":"Match\n\n$$f(x)=2(x+2)^2-7$$\n\nwith\n\n$$f(x)=a(x+b)^2+c.$$\n\nSo\n\n$$a=2,\\quad b=2,\\quad c=-7.$$\n\nGiving all three constants earns the final accuracy mark (A1).","highlights":[{"text":"$$f(x) = a(x + b)^2 + c$","location":"specification"}]}]}],"generatedAt":"2026-07-21T02:54:18.528Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":71}},{"id":"8c5e576e-aebd-4d00-a341-9092c3972611","revisionId":"7d8a80d1-c8d3-4cd5-b8dc-6fd787c4b852","questionNumber":"15","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":6,"sourceTitle":"Question Set 2: Quadratics","calculatorUse":null,"parts":[{"id":"afe6a63f-b9d9-4d44-83eb-0adbc5d5c21c","label":"a","content":"By completing the square, find in terms of the constant $k$ the roots of the equation\n\n$$x^2 + kx - 6 = 0$$","markscheme":"- $(x + \\frac{k}{2})^2 - \\frac{k^2}{4} - 6 = 0$. M1\n- $(x + \\frac{k}{2})^2 = \\frac{k^2}{4} + 6$. M1\n- $x + \\frac{k}{2} = \\pm \\sqrt{\\frac{k^2}{4} + 6}$. M1\n- $x = -\\frac{k}{2} \\pm \\sqrt{\\frac{k^2}{4} + 6}$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Completes the square correctly.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges to isolate the squared term.","code":"M"},{"id":"p3","marks":1,"description":"Takes the square root of both sides.","code":"M"},{"id":"p4","marks":1,"description":"Final answer $x = -\\frac{k}{2} \\pm \\sqrt{\\frac{k^2}{4} + 6}$.","code":"A"}],"finalAnswers":["x = -\\frac{k}{2} \\pm \\sqrt{\\frac{k^2}{4} + 6}"],"gener

Quadratics

Question 13

Quadratics Questions

Quadratics

Question 13

Quadratics Questions