a-7a22-4add-b915-ad79b18324ec","revisionId":"224be7a2-e431-4d29-a1c2-b58859ea7c33","questionNumber":"13","questionType":"LA","specification":"","difficulty":4,"difficultyBand":"hard","totalMarks":5,"sourceTitle":"Completing the Square","calculatorUse":"non_calc","parts":[{"id":"984ccc77-e1ed-4440-87cd-0ae383c438d2","label":"0","content":"By completing the square, solve $x^2 + 10x - 3 = 0$\nGive your answers in surd form.","markscheme":"- $(x + 5)^2 - 25 - 3 = 0$ M1\n- $(x + 5)^2 = 28$ M1\n- $x + 5 = \\pm\\sqrt{28}$ M1\n- $x = -5 \\pm \\sqrt{28}$ M1\n- $x = -5 \\pm 2\\sqrt{7}$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correctly completes the square to $(x + 5)^2 - 28 = 0$.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges to $(x + 5)^2 = 28$.","code":"M"},{"id":"p3","marks":1,"description":"Takes the square root of both sides, including $\\pm$.","code":"M"},{"id":"p4","marks":1,"description":"Isolates $x$ to get $x = -5 \\pm \\sqrt{28}$.","code":"M"},{"id":"p5","marks":1,"description":"Simplifies the surd to $2\\sqrt{7}$ and gives final answers.","code":"A"}],"finalAnswers":["-5 + 2\\sqrt{7}","-5 - 2\\sqrt{7}"],"generatedAt":"2026-05-06T11:14:00.474Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"completing_the_square","label":"Complete the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+10x-3=0$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=completed}{(x+5)^2-25}}-3=0$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"completed"}]},"order":0,"title":"Complete the square","description":"Half the coefficient of x to get 5. Squaring x + 5 introduces an extra 25, so subtract 25 to keep the expression unchanged. This earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+5)^2-28=0$"},{"latex":"$$(x+5)^2=\\color{@orange}{\\htmlData{anchor=isolated}{28}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"isolated"}]},"order":1,"title":"Isolate the square","description":"Combine the two negative constants, then add 28 to both sides. Reaching the displayed equation earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+5)^2=28$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=roots}{x+5=\\pm\\sqrt{28}}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"roots"}]},"order":2,"title":"Take square roots","description":"A square has both a positive and a negative square root, so include the plus-or-minus sign. This earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x+5=\\pm\\sqrt{28}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=subject}{x=-5\\pm\\sqrt{28}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"subject"}]},"order":3,"title":"Make x the subject","description":"Subtract 5 from both sides, keeping both signs. The resulting expression for x earns the fourth M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=-5\\pm\\sqrt{28}$"},{"latex":"$$x=-5\\pm\\sqrt{\\htmlData{anchor=factor}{4\\times7}}$"},{"latex":"$$\\color{@teal}{\\htmlData{anchor=answer}{x=-5\\pm2\\sqrt7}}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"factor"},{"color":"teal","style":"text-color-with-background","anchor":"answer"}]},"order":4,"title":"Simplify the surd","description":"Use the square factor 4 inside 28, then take its square root. The fully simplified surd answers earn the final A1."}]}],"generatedAt":"2026-07-19T15:35:00.021Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"984ccc77-e1ed-4440-87cd-0ae383c438d2:direct_completing_square","label":"Direct Completing the Square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2 + \\htmlData{anchor=coeff}{\\color{@sky}{10}}x - 3 = 0$"},{"latex":"$$(x + \\htmlData{anchor=half-coeff}{\\color{@sky}{5}})^2 - \\htmlData{anchor=square-term}{\\color{@amber}{5^2}} - 3 = 0$"}],"highl

Completing the Square

Question 8

Completing the Square Questions

Completing the Square

Question 8

Completing the Square Questions