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square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=original_terms}{x^2-11x}}+23.25=0$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=completed_terms}{(x-5.5)^2-30.25}}+23.25=0$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"original_terms"},{"color":"sky","style":"text-color","anchor":"completed_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,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/de1c1dc4-3f3d-4bf7-b1ff-42b209dd8441.mp4","durationSeconds":151,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/de1c1dc4-3f3d-4bf7-b1ff-42b209dd8441.jpg"}},{"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","isCalculation":true,"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"},"tableImages":[],"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 displayed line, then add 18 to both sides to isolate the square. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+2)^2=18$"},{"latex":"$$x+2=\\pm\\sqrt{18}$"}]},"order":2,"title":"Take square roots","description":"Take the square root of both sides. Both signs are needed because a positive and a negative value can have the same square. This result earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x+2=\\pm\\sqrt{18}$"},{"latex":"$$x=-2\\pm\\sqrt{18}$"}]},"order":3,"title":"Rearrange for x","description":"Subtract 2 from both sides to make x the subject. This earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=-2\\pm\\sqrt{18}$"},{"latex":"$$x=-2\\pm\\sqrt{9\\times2}$"},{"latex":"$$x=-2\\pm3\\sqrt{2}$"}]},"order":4,"title":"Simplify the surd","description":"Factorise 18 as 9 multiplied by 2, then use the fact that the square root of 9 is 3. This gives both exact solutions in the required surd form and earns A1."}]}],"generatedAt":"2026-07-19T02:13:30.278Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0ccc68d7-a51b-4492-9b11-3f012aad841d:completing_square","label":"Completing the Square","steps":[{"type":"text","order":0,"title":"Identify the method","content":"We are told to solve the quadratic by completing the square. That means rewriting $x^2+4x-14$ using a squared bracket.","highlights":[{"text":"By completing the square","location":"specification"},{"text":"Give your answers in surd form.","location":"specification"}]},{"type":"text","order":1,"title":"Complete the square","content":"For $x^2+4x$, halve the coefficient of $x$: half of $4$ is $2$. So:\n\n$$x^2+4x=(x+2)^2-4$$\n\nTherefore:\n\n$$x^2+4x-14=0$$\n\nbecomes\n\n$$(x+2)^2-4-14=0$$\n\nThis matches the first markscheme step and earns the first method mark."},{"type":"text","order":2,"title":"Simplify the equation","content":"Now combine the constant terms:\n\n$$-4-14=-18$$\n\nSo:\n\n$$(x+2)^2-18=0$$\n\nAdd $18$ to both sides:\n\n$$(x+2)^2=18$$\n\nThis earns the next method mark."},{"type":"text","order":3,"title":"Take square roots","content":"To remove the square, take the square root of both sides. Remember to include the $\\pm$ sign because both positive and negative values can square to $18$:\n\n$$x+2=\\pm\\sqrt{18}$$\n\nThis is another markscheme step."},{"type":"text","order":4,"title":"Solve for x","content":"Subtract $2$ from both sides:\n\n$$x=-2\\pm\\sqrt{18}$$\n\nSo the two solutions are currently:\n\n$$x=-2+\\sqrt{18} \\quad \\text{or} \\quad x=-2-\\sqrt{18}$$"},{"type":"text","order":5,"title":"Simplify the surd","content":"Now simplify $\\sqrt{18}$:\n\n$$\\sqrt{18}=\\sqrt{9\\times2}=3\\sqrt{2}$$\n\nSo:\n\n$$x=-2\\pm3\\sqrt{2}$$"},{"type":"text","order":6,"title":"Final answer","content":"The solutions in surd form are:\n\n$$\\boxed{x=-2+3\\sqrt{2}} \\quad \\text{or} \\quad \\boxed{x=-2-3\\sqrt{2}}$$\n\nThis earns the final accuracy mark because the surd has been simplified correctly."}]}],"generatedAt":"2026-06-08T12:37:26.341Z","modelVersion":"gpt-5.5","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/334eb85d-cc3c-48b4-8486-1bcdda24fa68.mp4","durationSeconds":96,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/334eb85d-cc3c-48b4-8486-1bcdda24fa68.jpg"}},{"id":"c8fc165f-d4cc-4878-984a-a8c930732ac4","revisionId":"35ca0e4a-25dd-4336-9e1b-d89da424cedd","questionNumber":null,"questionType":"LA","specification":"","difficulty":4,"difficultyBand":"hard","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"3b059758-2418-47fa-bac2-f58d24c66717","label":"a","content":"By completing the square solve $x^2 + 15x + 46.25 = 0$\nGive your answers in surd form.","markscheme":"- $(x + 7.5)^2 - 56.25 + 46.25 = 0$ M1\n- $(x + 7.5)^2 = 10$ M1\n- $x + 7.5 = \\pm\\sqrt{10}$ M1\n- $x = -7.5 \\pm \\sqrt{10}$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correct attempt to complete the square.","code":"M","acceptable":["(x + 7.5)^2 - 56.25 + 46.25 = 0","(x + 7.5)^2 - 10 = 0"]},{"id":"p2","marks":1,"description":"Isolating the squared bracket.","code":"M","acceptable":["(x + 7.5)^2 = 10"]},{"id":"p3","marks":1,"description":"Correct square rooting of both sides.","code":"M","acceptable":["x + 7.5 = \\pm\\sqrt{10}"]},{"id":"p4","marks":2,"description":"Both solutions correctly simplified and stated in surd form.","code":"A","acceptable":["x = -7.5 + \\sqrt{10}, x = -7.5 - \\sqrt{10}","x = -\\frac{15}{2} \\pm \\sqrt{10}"]}],"finalAnswers":["-7.5 \\pm \\sqrt{10}"],"generatedAt":"2026-05-06T14:07:19.232Z","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$: $15\\div 2=7.5$. The square will therefore begin with $(x+7.5)^2$.","highlights":[{"text":"By completing the square solve","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+15x+46.25=0$"},{"latex":"$$(\\htmlData{anchor=square}{\\color{@sky}{x+7.5}})^2-\\htmlData{anchor=added}{\\color{@amber}{56.25}}+46.25=0$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"square"},{"color":"amber","style":"text-color","anchor":"added"}]},"order":1,"title":"Complete the square","description":"Half the coefficient of x is 7.5. Add and subtract 7.5 squared, which is 56.25, inside the equation so the expression is unchanged. This earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=leftsquare}{(x+7.5)^2}}-56.25+46.25=0$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=isolated}{(x+7.5)^2}}=10$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"leftsquare"},{"color":"sky","style":"text-color-with-background","anchor":"isolated"}]},"order":2,"title":"Isolate the square","description":"Combine the constant terms and move the result to the other side. Writing this line correctly earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=square}{\\color{@sky}{(x+7.5)^2}}=10$"},{"latex":"$$\\htmlData{anchor=result}{\\color{@orange}{x+7.5=\\pm\\sqrt{10}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"square"},{"color":"orange","style":"text-color","anchor":"result"}]},"order":3,"title":"Take both square roots","description":"A square can come from either a positive or negative square root, so include both possibilities. This earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=lhs}{x+7.5}}=\\pm\\sqrt{10}$"},{"latex":"$$x=-7.5\\pm\\sqrt{10}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"lhs"}]},"order":4,"title":"State both solutions","description":"Subtract 7.5 from both sides. Keeping the plus-or-minus sign gives both roots in the required surd form, earning the final 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Completing the Square

Question 6

Completing the Square Questions

Completing the Square

Question 6

Completing the Square Questions