terms"}]},"order":1,"title":"Complete the square","description":"Half of −11 is −5.5, and 5.5² = 30.25. Since expanding (x − 5.5)² introduces +30.25, subtract 30.25 to keep the expression unchanged. This earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x-5.5)^2-30.25+23.25=0$"},{"latex":"$$(x-5.5)^2-\\htmlData{anchor=seven_left}{\\color{@green}{7}}=0$"},{"latex":"$$(x-5.5)^2=\\htmlData{anchor=seven_right}{\\color{@green}{7}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"seven_left"},{"color":"green","style":"text-color","anchor":"seven_right"}]},"order":2,"title":"Isolate the square","description":"Combine the constants, then add 7 to both sides. Reaching the isolated square earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x-5.5)^2=\\htmlData{anchor=root_source}{\\color{@purple}{7}}$"},{"latex":"$$x-5.5=\\htmlData{anchor=root_result}{\\color{@purple}{\\pm\\sqrt7}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"root_source"},{"color":"purple","style":"text-color","anchor":"root_result"}],"connections":[{"to":"root_result","from":"root_source","color":"purple","label":"square root both sides"}]},"order":3,"title":"Take square roots","description":"Taking square roots produces two possibilities, so the plus-or-minus sign is essential. This earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x-\\htmlData{anchor=five_left}{\\color{@green}{5.5}}=\\pm\\sqrt{7}$"},{"latex":"$$x=\\htmlData{anchor=five_right}{\\color{@green}{5.5}}\\pm\\sqrt{7}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"five_left"},{"color":"green","style":"text-color","anchor":"five_right"}]},"order":4,"title":"Solve for x","description":"Add 5.5 to both sides. The two exact answers in surd form earn the final A2."}]}],"generatedAt":"2026-07-19T07:23:02.520Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"334eb85d-cc3c-48b4-8486-1bcdda24fa68","revisionId":"8af0821e-9661-4e0b-85ab-9d6c15a01872","questionNumber":null,"questionType":"LA","specification":"","difficulty":4,"difficultyBand":"hard","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"0ccc68d7-a51b-4492-9b11-3f012aad841d","label":"a","content":"By completing the square, solve $x^2 + 4x - 14 = 0$\nGive your answers in surd form.","markscheme":"- $(x + 2)^2 - 4 - 14 = 0$ M1\n- $(x + 2)^2 = 18$ M1\n- $x + 2 = \\pm\\sqrt{18}$ M1\n- $x = -2 \\pm \\sqrt{18}$ M1\n- $x = -2 \\pm 3\\sqrt{2}$ 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":["-2 \\pm 3\\sqrt{2}"],"generatedAt":"2026-05-06T14:07:16.610Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"completing_the_square","label":"Completing the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+4x-14=0$"},{"latex":"$$(x+2)^2-4-14=0$"}]},"order":0,"title":"Complete the square","description":"Half of 4 is 2, so use the square x plus 2, squared. This introduces an extra 4, so subtract 4 to keep the expression equivalent. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+2)^2-4-14=0$"},{"latex":"$$(x+2)^2-18=0$"},{"latex":"$$(x+2)^2=18$"}]},"order":1,"title":"Isolate the square","description":"From the completed-square equation, simplify the constants to get the first d

Completing the Square

Question 5

Completing the Square Questions

Completing the Square

Question 5

Completing the Square Questions