:"hyper_latex","block":{"nodes":[{"latex":"$$\\left(x+3.5\\right)^2+\\color{@green}{\\htmlData{anchor=constants}{\\left(-12.25+9.25\\right)}}=0$"},{"latex":"$$\\left(x+3.5\\right)^2\\color{@green}{\\htmlData{anchor=minus_three}{-3}}=0$"},{"latex":"$$\\left(x+3.5\\right)^2=\\color{@teal}{\\htmlData{anchor=three}{3}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"constants"},{"color":"green","style":"text-color-with-background","anchor":"minus_three"},{"color":"teal","style":"text-color-with-background","anchor":"three"}]},"order":2,"title":"Isolate the square","description":"First combine the constant terms. Then add three to both sides; reaching the isolated square earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@purple}{\\htmlData{anchor=squared}{\\left(x+3.5\\right)^2}}=3$"},{"latex":"$$x+3.5=\\color{@purple}{\\htmlData{anchor=roots}{\\pm\\sqrt{3}}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"squared"},{"color":"purple","style":"text-color-with-background","anchor":"roots"}]},"order":3,"title":"Take both roots","description":"Taking a square root gives both a positive and a negative possibility. Including both signs earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x+\\color{@red}{\\htmlData{anchor=positive}{3.5}}=\\pm\\sqrt{3}$"},{"latex":"$$x=\\color{@red}{\\htmlData{anchor=negative}{-3.5}}\\pm\\sqrt{3}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"positive"},{"color":"red","style":"text-color-with-background","anchor":"negative"}]},"order":4,"title":"Solve for x","description":"Subtract three point five from both sides. The two exact answers in surd form earn the final A2."}]}],"generatedAt":"2026-07-19T08:38:15.329Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":61}},{"id":"107e1b9e-5d8e-4e13-8fbe-790c772c3287","revisionId":"86c984fc-daf8-45ff-9b32-f07abf9b1bb3","questionNumber":null,"questionType":"LA","specification":"","difficulty":4,"difficultyBand":"hard","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"9b104d35-dbd4-4153-9d1c-c99f1556b52a","label":"a","content":"By completing the square, solve $x^2 - 14x + 1 = 0$\nGive your answers in surd form.","markscheme":"- $(x - 7)^2 - 49 + 1 = 0$ M1\n- $(x - 7)^2 = 48$ M1\n- $x - 7 = \\pm\\sqrt{48}$ M1\n- $x = 7 \\pm \\sqrt{48}$ M1\n- $x = 7 \\pm 4\\sqrt{3}$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correctly completes the square for the quadratic expression.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges the equation to isolate the squared term.","code":"M"},{"id":"p3","marks":1,"description":"Takes the square root of both sides, including both positive and negative roots.","code":"M"},{"id":"p4","marks":1,"description":"Isolates $x$ to find the solutions in unsimplified surd form.","code":"M"},{"id":"p5","marks":1,"description":"Simplifies the surd to give final answer in the form $a \\pm b\\sqrt{c}$.","code":"A"}],"finalAnswers":["7 \\pm 4\\sqrt{3}"],"generatedAt":"2026-05-06T14:07:14.810Z","modelVersion":"gemini-3-flash-preview"},"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 $-14$ is $-7$, so the square will begin $(x-7)^2$.","highlights":[{"text":"By completing the square","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=equation_before}{\\color{@orange}{x^2-14x}+1=0}$"},{"latex":"$$\\htmlData{anchor=equation_after}{\\color{@sky}{(x-7)^2-49}+1=0}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"equation_before"},{"color":"sky","style":"text-color-with-background","anchor":"equation_after"}],"connections":[{"to":"equation_after","from":"equat

Completing the Square

Question 3

Completing the Square Questions

Completing the Square

Question 3

Completing the Square Questions