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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. 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":66},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/e1e3cbe8-237a-41a8-8f2d-0e135f09e424.mp4","durationSeconds":131,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/e1e3cbe8-237a-41a8-8f2d-0e135f09e424.jpg"}},{"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","isCalculation":true,"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"},"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 $x$. 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":"equation_before","color":"sky","label":"complete the square"}]},"order":1,"title":"Complete the square","description":"Expanding $(x-7)^2$ creates an extra $49$, so subtract $49$ to keep the expression unchanged. Writing this completed-square form earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=isolate_before}{(x-7)^2-\\color{@amber}{48}=0}$"},{"latex":"$$\\htmlData{anchor=isolate_after}{(x-7)^2=\\color{@amber}{48}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"isolate_before"},{"color":"amber","style":"text-color-with-background","anchor":"isolate_after"}],"connections":[{"to":"isolate_after","from":"isolate_before","color":"amber","label":"add 48 to both sides"}]},"order":2,"title":"Isolate the square","description":"Combine the constants, then add $48$ to both sides. Reaching $(x-7)^2=48$ earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=root_before}{\\color{@green}{(x-7)^2}=48}$"},{"latex":"$$\\htmlData{anchor=root_after}{\\color{@sky}{x-7}=\\pm\\sqrt{48}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"root_before"},{"color":"sky","style":"text-color-with-background","anchor":"root_after"}],"connections":[{"to":"root_after","from":"root_before","color":"sky","label":"square root both sides"}]},"order":3,"title":"Take square roots","description":"Taking square roots gives two possibilities, so the $\\pm$ sign is essential. This line earns the third M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=subject_before}{\\color{@amber}{x-7}=\\pm\\sqrt{48}}$"},{"latex":"$$\\htmlData{anchor=subject_after}{\\color{@amber}{x}=7\\pm\\sqrt{48}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"subject_before"},{"color":"amber","style":"text-color-with-background","anchor":"subject_after"}],"connections":[{"to":"subject_after","from":"subject_before","color":"amber","label":"add 7 to both sides"}]},"order":4,"title":"Make x the subject","description":"Add $7$ to both sides. This earns the fourth M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=7\\pm\\sqrt{\\htmlData{anchor=factor}{\\color{@purple}{16\\times3}}}$"},{"latex":"$$x=7\\pm\\htmlData{anchor=simplified}{\\color{@purple}{4\\sqrt3}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"factor"},{"color":"purple","style":"text-color","anchor":"simplified"}],"connections":[{"to":"simplified","from":"factor","color":"purple","label":"square root of 16 is 4"}]},"order":5,"title":"Simplify the surd","description":"Take the square factor $16$ out of the square root. Giving the fully simplified surd earns the final A1 mark."}]}],"generatedAt":"2026-07-19T19:00:40.932Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9b104d35-dbd4-4153-9d1c-c99f1556b52a:completing_the_square","label":"Completing the Square","steps":[{"type":"text","order":0,"title":"Complete the square","content":"To solve the quadratic equation $x^2 - 14x + 1 = 0$ by completing the square, we first focus on the $x^2 - 14x$ terms.\n\nWe halve the coefficient of the $x$ term:\n$$\\frac{-14}{2} = -7$$\n\nNow we write the expression as a perfect square and subtract the square of this value:\n$$x^2 - 14x = (x - 7)^2 - (-7)^2 = (x - 7)^2 - 49$$\n\nSubstitute this back into our original equation:\n$$(x - 7)^2 - 49 + 1 = 0$$\n\nThis earns our first mark **(M1)**.","highlights":[{"text":"By completing the square, solve $x^2 - 14x + 1 = 0$","location":"specification"}]},{"type":"text","order":1,"title":"Simplify and isolate","content":"Next, let's simplify the constant terms outside the bracket:\n$$-49 + 1 = -48$$\n\nSo the equation becomes:\n$$(x - 7)^2 - 48 = 0$$\n\nNow, add $48$ to both sides of the equation to isolate the squared bracket:\n$$(x - 7)^2 = 48$$\n\nThis earns the second mark **(M1)**."},{"type":"text","order":2,"title":"Take the square root","content":"To eliminate the square on the left-hand side, we take the square root of both sides. Remember to include both the positive and negative square roots:\n$$x - 7 = \\pm\\sqrt{48}$$\n\nThis step earns the third mark **(M1)**."},{"type":"text","order":3,"title":"Isolate x","content":"Add $7$ to both sides of the equation to solve for $x$:\n$$x = 7 \\pm \\sqrt{48}$$\n\nThis step earns the fourth mark **(M1)**."},{"type":"text","order":4,"title":"Simplify the surd","content":"To get our final answer in its simplest surd form, we need to simplify $\\sqrt{48}$. \n\nWe look for the largest square number that divides into $48$, which is $16$:\n$$\\sqrt{48} = \\sqrt{16 \\times 3} = \\sqrt{16} \\times \\sqrt{3} = 4\\sqrt{3}$$\n\nSubstituting this back gives our final solutions:\n$$x = 7 \\pm 4\\sqrt{3}$$\n\nThis final exact simplification earns the final accuracy mark **(A1)**.","highlights":[{"text":"Give your answers in surd form.","location":"specification"}]}]}],"generatedAt":"2026-06-12T10:07:42.540Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":67},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/107e1b9e-5d8e-4e13-8fbe-790c772c3287.mp4","durationSeconds":122,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/107e1b9e-5d8e-4e13-8fbe-790c772c3287.jpg"}},{"id":"5fbf5e43-d5a9-4882-86e1-b637ad11aeb3","revisionId":"0bcecd4d-22bc-4701-aa90-02dbf88c0b54","questionNumber":null,"questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"bd2daccc-2a92-4f3b-8d35-35c61bcc097f","label":"a","content":"Write $2x^2 + 20x + 42$ in the form $a(x + b)^2 + c$ where $a$, $b$, and $c$ are integers.","markscheme":"- $2(x^2 + 10x + 21)$ M1\n- $2((x + 5)^2 - 25 + 21)$ M1\n- $2(x + 5)^2 - 8$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Factorise 2 from the quadratic or x terms.","code":"M"},{"id":"p2","marks":1,"description":"Complete the square for the resulting $(x^2 + 10x)$ term.","code":"M"},{"id":"p3","marks":1,"description":"Obtain $2(x + 5)^2 - 8$.","code":"A"}],"finalAnswers":["2(x + 5)^2 - 8"],"generatedAt":"2026-05-06T14:07:04.790Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"completing_the_square","label":"Complete the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=original}{2x^2+20x+42}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=factored}{2(x^2+10x+21)}}$"}],"connections":[{"to":"factored","from":"original","color":"sky","label":"factor out 2"}]},"order":0,"title":"Factor out two","description":"First take the common factor from every term. This earns the first M1 method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2(x^2+\\color{@orange}{10x}+21)$"},{"latex":"$$2((x+\\color{@orange}{5})^2-\\color{@orange}{25}+21)$"}]},"order":1,"title":"Balance the square","description":"Half of 10 is 5, so use (x + 5)². This introduces 5² = 25, so subtract 25 inside the bracket. Showing this earns the second M1 method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2((x+5)^2-25+21)$"},{"latex":"$$2((x+5)^2+\\color{@green}{\\htmlData{anchor=minusfour}{(-4)}})$"},{"latex":"$$2(x+5)^2\\color{@green}{\\htmlData{anchor=minuseight}{-8}}$"}],"connections":[{"to":"minuseight","from":"minusfour","color":"green","label":"multiply by 2"}]},"order":2,"title":"Simplify the constant","description":"First combine the constants inside the bracket, then multiply −4 by the outer factor. The final form is 2(x + 5)² − 8, so a = 2, b = 5 and c = −8. This correct final form earns A1."}]}],"generatedAt":"2026-07-19T16:46:21.242Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"633f22d9-d346-4219-846a-867c58fa2ef7","label":"b","content":"Hence, or otherwise, write down the coordinates of the turning point of the graph of $y = 2x^2 + 20x + 42$","markscheme":"- $(-5, -8)$ B1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Correct coordinates $(-5, -8)$.","code":"B"}],"finalAnswers":["(-5, -8)"],"generatedAt":"2026-05-06T14:07:04.790Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"complete_the_square","label":"Complete the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=2x^2+20x+42$"},{"latex":"$$y=2\\left(\\htmlData{anchor=quadratic}{\\color{@sky}{x^2+10x}}+21\\right)$"},{"latex":"$$y=2\\left(\\htmlData{anchor=square}{\\color{@sky}{(x+5)^2-25}}+21\\right)$"},{"latex":"$$y=2(x+5)^2-8$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"quadratic"},{"color":"sky","style":"text-color","anchor":"square"}]},"order":1,"title":"Complete the square","description":"Completing the square rewrites the equation in turning-point form. First factor out 2, then halve the coefficient of x: half of 10 is 5, so subtract 5 squared to keep the expression unchanged."},{"type":"text","order":2,"title":"Read the coordinates","content":"The squared term is smallest when $x+5=0$, so $x=-5$. The corresponding value of $y$ is $-8$. Therefore the turning point is\n\n$$\\boxed{(-5,-8)}$$\n\nWriting this correct coordinate pair earns the **B1** mark; no working is required because the question says “write down”.","highlights":[{"text":"write down the coordinates","location":"specification"},{"text":"$$y = 2x^2 + 20x + 42$","location":"specification"}]}]}],"generatedAt":"2026-07-19T16:51:28.266Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"bd2daccc-2a92-4f3b-8d35-35c61bcc097f:factoring_all","label":"(a) Factoring Entire Expression First","steps":[{"type":"text","order":0,"title":"Factor out the coefficient","content":"To complete the square using the **Factoring Entire Expression First** method, we start by factoring out the coefficient of $x^2$, which is $2$, from the entire expression:\n\n$$2x^2 + 20x + 42 = 2(x^2 + 10x + 21)$$\n\nThis leaves us with a simpler quadratic expression, $x^2 + 10x + 21$, inside the parentheses.\n\n*This step earns the first method mark **M1**.*","highlights":[{"text":"2x^2 + 20x + 42","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{For } x^2 + \text{10}x + 21, \\text{ take half of the } x \\text{-coefficient: } \\frac{10}{2} = \\htmlData{anchor=half-coeff}{\\color{@sky}{5}}$"},{"latex":"$$x^2 + 10x + 21 = (x + \\htmlData{anchor=b-val}{\\color{@sky}{5}})^2 - \\htmlData{anchor=b-sq}{\\color{@sky}{5}}^2 + 21$"},{"latex":"$$\\phantom{x^2 + 10x + 21} = (x + 5)^2 - \\htmlData{anchor=const-val}{\\color{@amber}{25}} + 21$"},{"latex":"$$\\phantom{x^2 + 10x + 21} = (x + 5)^2 - \\htmlData{anchor=simplified-const}{\\color{@amber}{4}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"half-coeff"},{"color":"sky","style":"text-color","anchor":"b-val"},{"color":"amber","style":"text-color","anchor":"const-val"},{"color":"amber","style":"text-color","anchor":"simplified-const"}],"connections":[{"to":"b-val","from":"half-coeff","color":"sky","label":"half coefficient","style":"solid"},{"to":"const-val","from":"b-sq","color":"amber","label":"square","style":"solid"}]},"order":1,"title":"Complete the square inside","description":"Now, we focus entirely on completing the square for the expression inside the parentheses: $x^2 + 10x + 21$."},{"type":"text","order":2,"title":"Substitute and distribute","content":"Now substitute the completed square form back into the factored expression:\n\n$$2(x^2 + 10x + 21) = 2\\left[(x + 5)^2 - 4\\right]$$\n\n*This intermediate step earns the second method mark **M1**.*\n\nFinally, distribute the outer $2$ back to both terms inside the square brackets:\n\n$$2(x + 5)^2 - 2(4) = 2(x + 5)^2 - 8$$\n\nOur final answer is in the form $a(x + b)^2 + c$, where $a = 2$, $b = 5$, and $c = -8$.\n\n*This final form earns the accuracy mark **A1**.*"}]},{"id":"bd2daccc-2a92-4f3b-8d35-35c61bcc097f:factoring_x_terms","label":"(a) Factoring x-terms Only","steps":[{"type":"text","order":0,"title":"Group and factor x-terms","content":"To complete the square using the **Factoring x-terms Only** method, we first focus on the terms containing $x$, which are $2x^2 + 20x$. We factor out the coefficient of $x^2$ (which is $2$) from only these two terms, leaving the constant $+42$ outside the parentheses:\n\n$$2(x^2 + 10x) + 42$$","highlights":[{"text":"2x^2 + 20x + 42","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2(x^2 + \\htmlData{anchor=coeff}{\\color{@sky}{10}}x) + 42$"},{"latex":"$$2\\left[\\left(x + \\htmlData{anchor=half-coeff}{\\color{@sky}{5}}\\right)^2 - \\htmlData{anchor=sq-val}{\\color{@amber}{25}}\\right] + 42$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"coeff"},{"color":"sky","style":"text-color","anchor":"half-coeff"},{"color":"amber","style":"text-color","anchor":"sq-val"}],"connections":[{"to":"half-coeff","from":"coeff","color":"sky","label":"half of 10","style":"solid"},{"to":"sq-val","from":"half-coeff","color":"amber","label":"5^2 = 25","style":"dashed"}]},"order":1,"title":"Complete the square","description":"Now, we complete the square inside the parentheses for $x^2 + 10x$. We take half of the coefficient of $x$ (which is $5$), write $(x + 5)^2$, and then subtract the square of this value ($5^2 = 25$)."},{"type":"text","order":2,"title":"Expand and simplify","content":"Next, we expand the outer brackets by multiplying the $2$ back through the terms inside the square brackets:\n\n$$\\begin{aligned} &2\\left[(x + 5)^2 - 25\\right] + 42 \\\\ &= 2(x + 5)^2 - 2(25) + 42 \\\\ &= 2(x + 5)^2 - 50 + 42 \\end{aligned}$$\n\nNow, we simplify the constant terms outside the parentheses:\n\n$$-50 + 42 = -8$$"},{"type":"text","order":3,"title":"State final answer","content":"Combining everything together gives our expression in the form $a(x + b)^2 + c$:\n\n$$2(x + 5)^2 - 8$$\n\nHere, the integers are $a = 2$, $b = 5$, and $c = -8$.","highlights":[{"text":"a(x + b)^2 + c","location":"specification"}]}]},{"id":"bd2daccc-2a92-4f3b-8d35-35c61bcc097f:equating_coefficients","label":"(a) Equating Coefficients","steps":[{"type":"text","order":0,"title":"Understand the goal","content":"We want to write the quadratic expression $2x^2 + 20x + 42$ in the vertex form:\n\n$$a(x + b)^2 + c$$\n\nTo do this using the **Equating Coefficients** method, we will expand this target form and compare its coefficients directly with our original expression to solve for $a$, $b$, and $c$.","highlights":[{"text":"2x^2 + 20x + 42","location":"specification"},{"text":"a(x + b)^2 + c","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Target Form: } a(x + b)^2 + c$"},{"latex":"$$= a(x^2 + 2bx + b^2) + c$"},{"latex":"$$= \\htmlData{anchor=quad-term}{\\color{@sky}{a}}x^2 + \\htmlData{anchor=lin-term}{\\color{@amber}{2ab}}x + (\\htmlData{anchor=const-term}{\\color{@purple}{ab^2 + c}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"quad-term"},{"color":"amber","style":"text-color","anchor":"lin-term"},{"color":"purple","style":"text-color","anchor":"const-term"}],"connections":[]},"order":1,"title":"Expand the target form","description":"First, expand the brackets of our target expression $a(x + b)^2 + c$ to get a standard quadratic form."},{"type":"text","order":2,"title":"Equate the coefficients","content":"Now we compare our expanded form, $ax^2 + 2abx + (ab^2 + c)$, with the original expression, $2x^2 + 20x + 42$.\n\nBy matching the corresponding terms, we set up a system of equations:\n\n1. **$x^2$ coefficient:**\n $$a = 2$$\n\n2. **$x$ coefficient:**\n $$2ab = 20$$\n\n3. **Constant term:**\n $$ab^2 + c = 42$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Since } \\htmlData{anchor=a-val}{\\color{@sky}{a = 2}}, \\text{ we substitute it into the second equation:}$"},{"latex":"$$2(\\htmlData{anchor=a-sub}{\\color{@sky}{2}})b = 20 \\implies 4b = 20 \\implies \\htmlData{anchor=b-val}{\\color{@amber}{b = 5}}$"},{"latex":"$$\\text{Now substitute both } \\color{@sky}{a} \\text{ and } \\color{@amber}{b} \\text{ into the constant equation:}$"},{"latex":"$$(\\htmlData{anchor=a-const-sub}{\\color{@sky}{2}})(\\htmlData{anchor=b-const-sub}{\\color{@amber}{5}})^2 + c = 42$"},{"latex":"$$2(25) + c = 42 \\implies 50 + c = 42 \\implies \\htmlData{anchor=c-val}{\\color{@purple}{c = -8}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"a-val"},{"color":"amber","style":"text-color","anchor":"b-val"},{"color":"purple","style":"text-color","anchor":"c-val"}],"connections":[{"to":"a-sub","from":"a-val","color":"sky","label":"substitute a","style":"solid"},{"to":"b-const-sub","from":"b-val","color":"amber","label":"substitute b","style":"solid"}]},"order":3,"title":"Solve for the parameters","description":"Substitute the known values step-by-step to find the remaining variables."},{"type":"text","order":4,"title":"Write final expression","content":"Substitute the values $a = 2$, $b = 5$, and $c = -8$ back into the target vertex form:\n\n$$2(x + 5)^2 - 8$$\n\nThis matches the final completed square form required by the markscheme."}]},{"id":"633f22d9-d346-4219-846a-867c58fa2ef7:vertex_form","label":"(b) Using Vertex Form","steps":[{"type":"text","order":0,"title":"Recall the vertex form","content":"From the previous part of the question, we have the quadratic expression written in vertex form:\n\n$$y = 2(x+5)^2 - 8$$\n\nThe general vertex form of a quadratic equation is:\n\n$$y = a(x-h)^2 + k$$\n\nwhere the coordinates of the turning point (the vertex) are given directly by $(h, k)$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = a(x - \\htmlData{anchor=h-gen}{\\color{@sky}{h}})^2 + \\htmlData{anchor=k-gen}{\\color{@pink}{k}}$"},{"latex":"$$y = 2(x - (\\htmlData{anchor=h-val}{\\color{@sky}{-5}}))^2 + (\\htmlData{anchor=k-val}{\\color{@pink}{-8}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"h-val"},{"color":"pink","style":"text-color","anchor":"k-val"}],"connections":[{"to":"h-val","from":"h-gen","color":"sky","label":null,"style":"solid"},{"to":"k-val","from":"k-gen","color":"pink","label":null,"style":"solid"}]},"order":1,"title":"Identify h and k","description":"To match our equation to the general form, we can rewrite addition as subtracting a negative, and subtraction as adding a negative."},{"type":"text","order":2,"title":"State the coordinates","content":"By comparing the two equations, we find:\n- $h = -5$\n- $k = -8$\n\nTherefore, the coordinates of the turning point are:\n\n$$\\mathbf{(-5, -8)}$$"}]},{"id":"633f22d9-d346-4219-846a-867c58fa2ef7:axis_of_symmetry","label":"(b) Using Axis of Symmetry Formula","steps":[{"type":"text","order":0,"title":"Identify the coefficients","content":"First, let's identify the coefficients of our quadratic equation:\n\n$$y = 2x^2 + 20x + 42$$\n\nA quadratic equation in standard form is written as $y = ax^2 + bx + c$. By comparing the equations, we have:\n- $a = 2$\n- $b = 20$\n- $c = 42$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=a-val}{\\color{@sky}{a = 2}}, \\quad \\htmlData{anchor=b-val}{\\color{@amber}{b = 20}}$"},{"latex":"$$x = -\\frac{\\htmlData{anchor=b-form}{\\color{@amber}{b}}}{2\\htmlData{anchor=a-form}{\\color{@sky}{a}}}$"},{"latex":"$$x = -\\frac{\\color{@amber}{20}}{2(\\color{@sky}{2})} = -5$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"b-val"},{"color":"sky","style":"text-color","anchor":"a-val"}],"connections":[{"to":"b-form","from":"b-val","color":"amber","label":null,"style":"solid"},{"to":"a-form","from":"a-val","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Find the x-coordinate","description":"The x-coordinate of the turning point lies on the axis of symmetry. We can find this using the formula:"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=x-val}{\\color{@purple}{x = -5}}$"},{"latex":"$$y = 2\\htmlData{anchor=x-sub1}{\\color{@purple}{x}}^2 + 20\\htmlData{anchor=x-sub2}{\\color{@purple}{x}} + 42$"},{"latex":"$$y = 2(\\color{@purple}{-5})^2 + 20(\\color{@purple}{-5}) + 42$"},{"latex":"$$y = 2(25) - 100 + 42 = -8$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"x-val"}],"connections":[{"to":"x-sub1","from":"x-val","color":"purple","label":null,"style":"solid"},{"to":"x-sub2","from":"x-val","color":"purple","label":null,"style":"solid"}]},"order":2,"title":"Find the y-coordinate","description":"Now, substitute this x-value back into the original quadratic equation to calculate the corresponding y-coordinate."},{"type":"text","order":3,"title":"State the turning point","content":"The coordinates of the turning point are written in the form $(x, y)$.\n\nUsing our calculated values:\n- $x = -5$\n- $y = -8$\n\nTherefore, the coordinates of the turning point are **$(-5, -8)$**."}]},{"id":"633f22d9-d346-4219-846a-867c58fa2ef7:differentiation","label":"(b) Using Differentiation","steps":[{"type":"text","order":0,"title":"Differentiate the equation","content":"To find the coordinates of the turning point using differentiation, we first find the derivative of the curve's equation. This gives us the gradient function.\n\nGiven the equation:\n$$y = 2x^2 + 20x + 42$$\n\nApplying the power rule for differentiation ($$\\frac{d}{dx}(x^n) = nx^{n-1}$$), we get:\n$$\\frac{dy}{dx} = 4x + 20$$","highlights":[{"text":"coordinates of the turning point of the graph of $y = 2x^2 + 20x + 42$","location":"specification"}]},{"type":"text","order":1,"title":"Find the x-coordinate","content":"At the turning point, the gradient of the curve is zero ($$\\frac{dy}{dx} = 0$$). Set the derivative to zero and solve for $$x$$:\n\n$$\\begin{aligned} 4x + 20 &= 0 \\\\ 4x &= -20 \\\\ x &= -5 \\end{aligned}$$"},{"type":"text","order":2,"title":"Find the y-coordinate","content":"Now, substitute $$x = -5$$ back into the original equation to find the corresponding $$y$$-coordinate:\n\n$$\\begin{aligned} y &= 2(-5)^2 + 20(-5) + 42 \\\\ y &= 2(25) - 100 + 42 \\\\ y &= 50 - 100 + 42 \\\\ y &= -8 \\end{aligned}$$\n\nThus, the coordinates of the turning point are $$(-5, -8)$$."}]}],"generatedAt":"2026-06-12T18:02:42.015Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":91},"walkthroughVideo":null}],"subjectCode":"gcse_maths__aqa","summaries":[{"id":"7f40db69-ca7c-4285-8e09-34656d81ce60","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"21a3243d-e573-401b-8dc0-efbb15dce586","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"45e23f29-00b9-4784-a691-c37252ca769a","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"4bba162f-d121-4811-9834-bd02bdc79c72","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"57d14b80-8895-4ab3-9945-f85df87adcf7","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"72ea4483-d396-4e36-babe-dff7fc59edf0","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8f3b2c7e-0db1-4544-b5f6-0958506eba55","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"92fa7c1c-24e6-44d8-9ee5-dc00117d9897","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"93d24245-b18f-40b3-9a32-cf4910080c1a","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a9a73ed0-e1d6-4198-a9c0-b1c6575eba3f","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b5046ff9-42b4-4eff-ae7a-612609e22512","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"baf0db88-ad54-4d69-b2cc-318e78918ad3","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c2839e7e-1bc5-4a2e-972a-69fbda186abc","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e1e3cbe8-237a-41a8-8f2d-0e135f09e424","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"fbe671f3-0591-4322-a71b-e8126e2b1d53","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"55526e85-0c25-45ef-95c8-9a6b4dcbed55","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"107e1b9e-5d8e-4e13-8fbe-790c772c3287","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"1b3db67d-ab4d-4d96-82c0-f3d48507bae5","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"29a74acf-02a7-4bcc-9f80-8a2fbbca1909","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5fbf5e43-d5a9-4882-86e1-b637ad11aeb3","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"728a5558-597a-4731-a3bd-a81de23bdddb","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"815514e4-689e-41a2-80f6-fe424fe892e6","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"837260c4-26bf-46ad-a682-db9fd32252b2","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9c7fb861-56b4-49c0-a030-4f732d65eeef","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9dc22c24-570d-4743-9966-a248e666473d","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b9b8c31b-60a5-4aac-a82f-32d5a29cf792","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"de1c1dc4-3f3d-4bf7-b1ff-42b209dd8441","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e6ac7d87-563f-48f1-836d-f2de17a1e004","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f07cdb42-421e-46ec-841a-9cbbb3689031","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"fb8fe8f6-dea3-47e3-9496-bcb0b03b6c30","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"af02dd24-49ac-42c1-9995-5ee875c053dc","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","is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the Square","topicId":"34f41bde-d2a1-4bbb-9d93-2f5b999a8342","topicSidebar":"$L3b","topicSlug":"grade-8-9-8-4-completing-the-square","topicTitle":"Completing the Square"}]}],"$L3c","$L3d"]}]

Completing the Square

Question 3

Completing the Square Questions

Completing the Square

Question 3

Completing the Square Questions