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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":["-4 \\pm 2\\sqrt{5}"],"generatedAt":"2026-05-06T14:07:12.105Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"completing_the_square","label":"Completing the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+8x-4=0$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=completed}{(x+4)^2-16}}-4=0$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"completed"}]},"order":1,"title":"Complete the square","description":"Half the coefficient of x, then square it. The bracket introduces an extra 16, so subtract 16 to keep the expression unchanged. This earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+4)^2-16-4=0$"},{"latex":"$$\\htmlData{anchor=square}{(x+4)^2}=\\color{@amber}{\\htmlData{anchor=twenty}{20}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"square"},{"color":"amber","style":"text-color-with-background","anchor":"twenty"}]},"order":2,"title":"Isolate the square","description":"Add 16 and 4 to both sides. Reaching the equation with 20 on the right earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x+4)^2=20$"},{"latex":"$$x+4=\\color{@sky}{\\htmlData{anchor=roots}{\\pm\\sqrt{20}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"roots"}]},"order":3,"title":"Take square roots","description":"Taking a square root gives two possibilities, so include the plus-or-minus sign. This earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x+4=\\pm\\sqrt{20}$"},{"latex":"$$\\htmlData{anchor=answer_unsimplified}{x=\\color{@amber}{-4}\\pm\\sqrt{20}}$"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"answer_unsimplified"}]},"order":4,"title":"Make x the subject","description":"Subtract 4 from both sides. This form earns the fourth M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=-4\\pm\\sqrt{20}$"},{"latex":"$$\\sqrt{20}=\\sqrt{\\color{@teal}{\\htmlData{anchor=factor}{4\\times5}}}$"},{"latex":"$$\\sqrt{20}=\\color{@teal}{\\htmlData{anchor=simplified}{2\\sqrt5}}$"},{"latex":"$$\\htmlData{anchor=final}{x=-4\\pm\\color{@teal}{2\\sqrt5}}$"}],"highlights":[{"color":"teal","style":"text-color-with-background","anchor":"factor"},{"color":"teal","style":"text-color-with-background","anchor":"simplified"},{"color":"teal","style":"text-color-with-background","anchor":"final"}]},"order":5,"title":"Simplify the surd","description":"Write 20 as 4 times 5, then take the square factor 4 out of the square root. Giving both exact answers in simplified surd form earns the final A1."}]}],"generatedAt":"2026-07-19T15:32:59.582Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"491158e7-3ace-4cdb-9887-c67a170800e6:balancing_equation","label":"Balancing Both Sides","steps":[{"type":"text","order":0,"title":"Move the constant term","content":"To begin completing the square using the balancing both sides approach, we start by moving the constant term to the right-hand side of the equation.\n\nGiven equation:\n$$x^2 + 8x - 4 = 0$$\n\nAdd $4$ to both sides to isolate the $x$-terms:\n$$x^2 + 8x = 4$$","highlights":[{"text":"x^2 + 8x - 4 = 0","location":"specification"}]},{"type":"text","order":1,"title":"Calculate term to add","content":"To create a perfect square trinomial on the left-hand side, we need to add the square of half the coefficient of $x$ to both sides.\n\nFirst, find the coefficient of $x$:\n$$\\text{Coefficient} = 8$$\n\nNow, take half of this value and square it:\n$$\\left(\\frac{8}{2}\\right)^2 = 4^2 = 16$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2 + 8x = 4$"},{"latex":"$$x^2 + 8x + \\htmlData{anchor=add-left}{\\color{@sky}{16}} = 4 + \\htmlData{anchor=add-right}{\\color{@sky}{16}}$"},{"latex":"$$x^2 + 8x + 16 = 20$"}],"highlights":[{"color":"sky","style":"box","anchor":"add-left"},{"color":"sky","style":"box","anchor":"add-right"}],"connections":[{"to":"add-right","from":"add-left","color":"sky","label":"add to both sides","style":"solid"}]},"order":2,"title":"Balance both sides","description":"Add $16$ to both sides of the equation to maintain equality. This earns you the first mark $(\\text{M1})$ for finding that $(x + 4)^2 = 20$."},{"type":"text","order":3,"title":"Write as perfect square","content":"Now, factor the left-hand side as a perfect square:\n$$(x + 4)^2 = 20$$\n\n*(This step earns another mark $\\text{M1}$)*."},{"type":"text","order":4,"title":"Solve for $x$","content":"Take the square root of both sides to begin solving for $x$:\n$$x + 4 = \\pm\\sqrt{20}$$ *(Earns mark $\\text{M1}$)*\n\nSubtract $4$ from both sides:\n$$x = -4 \\pm \\sqrt{20}$$ *(Earns mark $\\text{M1}$)*"},{"type":"text","order":5,"title":"Simplify the surd","content":"To get our final answer in simplified surd form, we simplify $\\sqrt{20}$ by identifying the largest square number factor of $20$:\n$$\\sqrt{20} = \\sqrt{4 \\times 5} = 2\\sqrt{5}$$\n\nSubstituting this back gives the final simplified solution:\n$$x = -4 \\pm 2\\sqrt{5}$$\n\n*(Earns final mark $\\text{A1}$)*","highlights":[{"text":"Give your answers in surd form.","location":"specification"}]}]},{"id":"491158e7-3ace-4cdb-9887-c67a170800e6:expression_rewriting","label":"Rewriting the Quadratic Expression","steps":[{"type":"text","order":0,"title":"Complete the square","content":"To solve the equation $x^2 + 8x - 4 = 0$ using the **Rewriting the Quadratic Expression** method, we start by looking at the first two terms: $x^2 + 8x$.\n\nTo write this in completed square form, we halve the coefficient of $x$ (which is $8$, giving us $4$) and write:\n$$(x + 4)^2$$\n\nSince $(x + 4)^2 = x^2 + 8x + 16$, we must subtract $16$ to keep the expression equivalent to $x^2 + 8x$:\n$$x^2 + 8x = (x + 4)^2 - 16$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\{ \\htmlData{anchor=orig-quad}{\\color{@sky}{x^2 + 8x}} \\} - 4 = 0$"},{"latex":"$$\\{ \\htmlData{anchor=comp-square}{\\color{@sky}{(x + 4)^2 - 16}} \\} - 4 = 0$"}],"highlights":[{"color":"sky","style":"box","anchor":"orig-quad"},{"color":"sky","style":"box","anchor":"comp-square"}],"connections":[{"to":"comp-square","from":"orig-quad","color":"sky","label":"Substitute","style":"solid"}]},"order":1,"title":"Substitute and rewrite","description":"Now, substitute $(x + 4)^2 - 16$ back into our original equation in place of $x^2 + 8x$."},{"type":"text","order":2,"title":"Simplify the equation","content":"Next, we simplify the constant terms on the left-hand side of the equation:\n\n$$\\begin{aligned}\n(x + 4)^2 - 16 - 4 &= 0 \\\\\n(x + 4)^2 - 20 &= 0\n\\end{aligned}$$\n\nThis earns the first method mark M1."},{"type":"text","order":3,"title":"Isolate the squared term","content":"To solve for $x$, we first isolate the squared term by adding $20$ to both sides of the equation:\n\n$$(x + 4)^2 = 20$$\n\nThis step earns the second method mark M1."},{"type":"text","order":4,"title":"Take the square root","content":"Take the square root of both sides to get rid of the exponent. Remember to include both the positive and negative square roots:\n\n$$x + 4 = \\pm\\sqrt{20}$$\n\nThis earns the third method mark M1."},{"type":"text","order":5,"title":"Solve for $x$","content":"Subtract $4$ from both sides to find the solutions for $x$:\n\n$$x = -4 \\pm \\sqrt{20}$$\n\nThis earns the fourth method mark M1."},{"type":"text","order":6,"title":"Simplify the surd","content":"To give the final answer in its simplest surd form, we simplify $\\sqrt{20}$ by finding the largest square factor of $20$, which is $4$:\n\n$$\\sqrt{20} = \\sqrt{4 \\times 5} = 2\\sqrt{5}$$\n\nSubstituting this back gives:\n\n$$x = -4 \\pm 2\\sqrt{5}$$\n\nThis final exact answer earns the accuracy mark A1."}]}],"generatedAt":"2026-06-12T23:11:58.100Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":35},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/93d24245-b18f-40b3-9a32-cf4910080c1a.mp4","durationSeconds":109,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/93d24245-b18f-40b3-9a32-cf4910080c1a.jpg"}},{"id":"e1e3cbe8-237a-41a8-8f2d-0e135f09e424","revisionId":"67d2e57f-be4e-4d4e-96f0-05f6e3ba7434","questionNumber":null,"questionType":"LA","specification":"","difficulty":4,"difficultyBand":"hard","totalMarks":5,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"e1af1ba7-7545-47ff-b6f3-b59731e36716","label":"a","content":"By completing the square solve $x^2 + 7x + 9.25 = 0$\nGive your answers in surd form.","markscheme":"- $(x + 3.5)^2 - 12.25 + 9.25 = 0$ M1\n- $(x + 3.5)^2 = 3$ M1\n- $x + 3.5 = \\pm\\sqrt{3}$ M1\n- $x = -3.5 \\pm \\sqrt{3}$ A2","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Correct attempt to complete the square showing the squared bracket and the correction term.","code":"M","acceptable":["(x + 3.5)^2 - 3.5^2 + 9.25 = 0","(x + 3.5)^2 - 3 = 0"]},{"id":"p2","marks":1,"description":"Isolating the squared bracket on one side of the equation.","code":"M","acceptable":["(x + 3.5)^2 = 3"]},{"id":"p3","marks":1,"description":"Taking the square root of both sides including the plus/minus sign.","code":"M","acceptable":["x + 3.5 = \\pm\\sqrt{3}"]},{"id":"p4","marks":2,"description":"Both solutions correctly simplified and stated in surd form.","code":"A","acceptable":["x = -3.5 + \\sqrt{3}, x = -3.5 - \\sqrt{3}","x = -\\frac{7}{2} \\pm \\sqrt{3}"]}],"finalAnswers":["-3.5 \\pm \\sqrt{3}"],"generatedAt":"2026-05-06T14:07:15.562Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"completing_the_square","label":"Completing the square","steps":[{"type":"text","order":0,"title":"Halve the coefficient","content":"To complete the square, halve the coefficient of $x$. Half of $7$ is $3.5$, so the squared bracket must begin $(x+3.5)^2$.","highlights":[{"text":"By completing the square","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+\\color{@sky}{\\htmlData{anchor=linear}{7x}}+9.25=0$"},{"latex":"$$\\left(x+\\color{@sky}{\\htmlData{anchor=half}{3.5}}\\right)^2-\\color{@orange}{\\htmlData{anchor=square}{(3.5)^2}}+9.25=0$"},{"latex":"$$\\left(x+3.5\\right)^2-\\color{@orange}{\\htmlData{anchor=value}{12.25}}+9.25=0$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear"},{"color":"sky","style":"text-color-with-background","anchor":"half"},{"color":"orange","style":"text-color","anchor":"square"},{"color":"orange","style":"text-color-with-background","anchor":"value"}]},"order":1,"title":"Complete the square","description":"Halve the coefficient of the linear term, then subtract the square you have introduced. This completed-square form earns the first M1."},{"type":"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. 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Completing the Square

Question 1

Completing the Square Questions

Completing the Square

Question 1

Completing the Square Questions