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bg-background","children":["$","$L3b",null,{"backHref":"/igcse/maths/cie/questions","basePath":"/igcse/maths/cie/grade-8-9-8-1-quadratic-simultaneous-equations/questions","formulaUrl":null,"initialIndex":6,"initialQuestionWindow":[{"id":"0cc49e8e-d89b-426d-ad24-b2f180fd7bba","revisionId":"3445ce4f-71ae-4a2a-a31c-aac486e40c91","questionNumber":"6","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","totalMarks":5,"sourceTitle":"Quadratic Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"6b5c4dff-11d3-4f86-adff-5a3c886c704f","label":"0","content":"Solve the simultaneous equations \nGive your answers to 3 significant figures\n\n$$x^2 + y^2 = 20$$\n\n$$2x + y = 3$$","markscheme":"- Rearranges linear to $y = 3 - 2x$ and substitutes: $x^2 + (3 - 2x)^2 = 20$ M1\n- Simplifies to $5x^2 - 12x - 11 = 0$ M1\n- Uses quadratic formula: $x = \\frac{12 \\pm \\sqrt{(-12)^2 - 4(5)(-11)}}{10}$ M1\n- Finds $x = 3.11$ and $x = -0.708$ A1\n- Finds $y = -3.22$ and $y = 4.42$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes linear into quadratic and expands brackets.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to a quadratic equation $5x^2 - 12x - 11 = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Applies the quadratic formula correctly.","code":"M"},{"id":"p4","marks":1,"description":"Finds both $x$ values to 3 significant figures.","code":"A"},{"id":"p5","marks":1,"description":"Finds both $y$ values to 3 significant figures.","code":"A"}],"finalAnswers":["x = 3.11, y = -3.22; x = -0.708, y = 4.42"],"generatedAt":"2026-05-07T02:12:13.942Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitution_quadratic_formula","label":"Substitution and quadratic formula","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@amber}{\\htmlData{anchor=linear}{2x+y=3}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=subject}{y=3-2x}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"linear"},{"color":"sky","style":"text-color","anchor":"subject"}],"connections":[{"to":"subject","from":"linear","color":"sky","label":"subtract 2x"}]},"order":0,"title":"Make y the subject","description":"Rearrange the linear equation so that it can be substituted into the circle equation."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^2+\\htmlData{anchor=original_y}{\\color{@orange}{y^2}}=20$"},{"latex":"$$x^2+\\htmlData{anchor=substituted_bracket}{\\color{@sky}{(3-2x)^2}}=20$"}]},"order":1,"title":"Substitute for y","description":"Replace y in the first equation with the whole expression 3 − 2x. Rearranging to y = 3 − 2x and then substituting to obtain x² + (3 − 2x)² = 20 together earn the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=source_equation}{x^2+(3-2x)^2=20}$"},{"latex":"$$\\htmlData{anchor=expanded_equation}{x^2+9-12x+4x^2=20}$"},{"latex":"$$\\htmlData{anchor=quadratic_equation}{5x^2-12x-11=0}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"source_equation"},{"color":"sky","style":"text-color","anchor":"expanded_equation"},{"color":"green","style":"text-color","anchor":"quadratic_equation"}]},"order":2,"title":"Form the quadratic","description":"First expand (3 − 2x)² to obtain x² + 9 − 12x + 4x² = 20. Then collect the x² terms and move everything to one side to obtain 5x² − 12x − 11 = 0. Obtaining this quadratic earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\dfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}$"},{"latex":"$$x=\\color{@sky}{\\htmlData{anchor=substitution}{\\dfrac{12\\pm\\sqrt{(-12)^2-4(5)(-11)}}{10}}}$"},{"latex":"$$x=\\color{@green}{\\htmlData{anchor=discriminant}{\\dfrac{12\\pm\\sqrt{364}}{10}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"green","style":"text-color","anchor":"discriminant"}],"connections":[{"to":"discriminant","from":"substitution","color":"sky","label":"evaluate the discriminant"}]},"order":3,"title":"Use the formula","description":"For 5x² − 12x − 11 = 0, use a = 5, b = −12 and c = −11. This correct formula setup earns the third M1."},{"type":"text","order":4,"title":"Round both x-values","content":"Evaluating both signs gives $x=3.107878\\ldots$ or $x=-0.707878\\ldots$. To **3 significant figures**, these are\n\n$$x=3.11 \\quad \\text{or} \\quad x=-0.708.$$\n\nBoth values are needed because the quadratic has two solutions. These correctly rounded $x$-values earn A1.","highlights":[{"text":"Give your answers to 3 significant figures","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@orange}{\\htmlData{anchor=xpositive}{x=3.107878\\ldots}}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=ynegative}{y=3-2(3.107878\\ldots)=-3.215756\\ldots}}$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=xnegative}{x=-0.707878\\ldots}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=ypositive}{y=3-2(-0.707878\\ldots)=4.415756\\ldots}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"xpositive"},{"color":"sky","style":"text-color","anchor":"ynegative"},{"color":"purple","style":"text-color","anchor":"xnegative"},{"color":"green","style":"text-color","anchor":"ypositive"}],"connections":[{"to":"ynegative","from":"xpositive","color":"sky","label":"substitute into y = 3 − 2x"},{"to":"ypositive","from":"xnegative","color":"green","label":"substitute into y = 3 − 2x"}]},"order":5,"title":"Find the matching y-values","description":"Substitute each unrounded x-value into y = 3 − 2x. Keep the pairs matched; rounding x too early can reduce accuracy."},{"type":"text","order":6,"title":"State the solution pairs","content":"Rounding each $y$-value to $3$ significant figures gives the two solutions\n\n$$\\boxed{(x,y)=(3.11,-3.22)}$$\n\nand\n\n$$\\boxed{(x,y)=(-0.708,4.42)}.$$\n\nWriting the matching $y$-values correctly earns the final A1."}]}],"generatedAt":"2026-07-20T16:53:22.734Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":43},"walkthroughVideo":null},{"id":"5a91a9ec-0d6f-4980-aa55-d2064da49327","revisionId":"8b9902b5-1ef5-44f7-b70d-3c9585611388","questionNumber":"7","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","totalMarks":5,"sourceTitle":"Quadratic Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"3e3cffef-7c48-4b8c-aa1b-21161d7c5bb6","label":"0","content":"Solve the simultaneous equations \nGive your answers to 3 significant figures\n\n$$x^2 + y^2 = 27$$\n\n$$2x - y = 3$$","markscheme":"- Rearranges linear to $y = 2x - 3$ and substitutes: $x^2 + (2x - 3)^2 = 27$ M1\n- Simplifies to $5x^2 - 12x - 18 = 0$ M1\n- Uses quadratic formula: $x = \\frac{12 \\pm \\sqrt{(-12)^2 - 4(5)(-18)}}{10}$ M1\n- Finds $x = 3.44$ and $x = -1.04$ A1\n- Finds $y = 3.89$ and $y = -5.09$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes linear into quadratic and expands brackets.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to a quadratic equation $5x^2 - 12x - 18 = 0$.","code":"M"},{"id":"p3","marks":1,"description":"Applies the quadratic formula correctly.","code":"M"},{"id":"p4","marks":1,"description":"Finds both $x$ values to 3 significant figures.","code":"A"},{"id":"p5","marks":1,"description":"Finds both $y$ values to 3 significant figures.","code":"A"}],"finalAnswers":["x = 3.44, y = 3.89; x = -1.04, y = -5.09"],"generatedAt":"2026-05-07T02:12:13.942Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitution_quadratic_formula","label":"Substitution and quadratic formula","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"2x-y=3"},{"latex":"y=\\htmlData{anchor=yexpr}{\\color{@sky}{2x-3}}"},{"latex":"27=x^2+\\htmlData{anchor=ysub}{\\color{@sky}{(2x-3)^2}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"yexpr"},{"color":"sky","style":"text-color","anchor":"ysub"}],"connections":[{"to":"ysub","from":"yexpr","color":"sky","label":"substitute for y","style":"solid"}]},"order":0,"title":"Rearrange and substitute","description":"Make $y$ the subject of the linear equation, then substitute that expression into the circle equation. This complete substitution earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(2x-3)^2=\\htmlData{anchor=expanded}{\\color{@orange}{4x^2-12x+9}}$"},{"latex":"$$27=x^2+\\htmlData{anchor=used}{\\color{@orange}{4x^2-12x+9}}$"},{"latex":"$$\\htmlData{anchor=quadratic}{\\color{@orange}{5x^2-12x-18}}=0$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"expanded"},{"color":"orange","style":"text-color-with-background","anchor":"used"},{"color":"orange","style":"text-color-with-background","anchor":"quadratic"}],"connections":[{"to":"used","from":"expanded","color":"orange","label":"substitute expansion"}]},"order":1,"title":"Form the quadratic","description":"Expand the square, substitute the expansion, and collect like terms. Reaching the simplified quadratic earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=\\dfrac{-b\\pm\\sqrt{b^2-4ac}}{2a}$"},{"latex":"$$x=\\dfrac{-(-12)\\pm\\sqrt{(-12)^2-4(5)(-18)}}{2(5)}$"},{"latex":"$$x=\\color{@green}{\\dfrac{12\\pm\\sqrt{(-12)^2-4(5)(-18)}}{10}}$"}]},"order":2,"title":"Apply the formula","description":"Use a = 5, b = -12 and c = -18 in the quadratic formula. A correct substitution earns the third M1 mark."},{"type":"text","order":3,"title":"Round both roots","content":"Evaluating the two signs gives $x=3.444994\\ldots$ or $x=-1.044994\\ldots$. To $3$ significant figures,\n\n$$x=3.44\\quad\\text{or}\\quad x=-1.04$$\n\nBoth roots are needed for the A1 mark.","highlights":[{"text":"Give your answers to 3 significant figures","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=2(3.444994\\ldots)-3$"},{"latex":"$$y=\\color{@teal}{3.889988\\ldots}$"},{"latex":"$$y=2(-1.044994\\ldots)-3$"},{"latex":"$$y=\\color{@purple}{-5.089988\\ldots}$"}]},"order":4,"title":"Find matching y-values","description":"Substitute each unrounded x-value into y = 2x - 3. Keep the pairs matched: the positive x gives the positive y, and the negative x gives the negative y."},{"type":"text","order":5,"title":"State both solutions","content":"Rounding each coordinate to $3$ significant figures, the simultaneous equations have two solutions:\n\n$$(x,y)=(3.44,\\,3.89)$$\n\nand\n\n$$(x,y)=(-1.04,\\,-5.09).$$\n\nGiving both correctly matched $y$-values earns the final A1 mark.","highlights":[{"text":"Give your answers to 3 significant figures","location":"specification"}]}]}],"generatedAt":"2026-07-20T16:50:56.899Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"05721abb-3610-49f7-aaa6-e71c8d3116a4","revisionId":"de9b912c-4257-4c05-bdc0-b650113622f1","questionNumber":"8","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","totalMarks":5,"sourceTitle":"Quadratic Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"241affb5-ca29-46b2-be62-fa9f96a63516","label":"0","content":"Solve algebraically the simultaneous equations\n\n$$x^2 - 3y^2 = 13$$\n\n$$2x + 3y = 4$$","markscheme":"- Rearranges linear to $x = \\frac{4 - 3y}{2}$ and substitutes: $(\\frac{4 - 3y}{2})^2 - 3y^2 = 13$ M1\n- Expands and clears denominator: $\\frac{16 - 24y + 9y^2}{4} - 3y^2 = 13 \\implies 16 - 24y + 9y^2 - 12y^2 = 52$ M1\n- Simplifies to $3y^2 + 24y + 36 = 0$ or $y^2 + 8y + 12 = 0$ M1\n- Factorises to $(y + 2)(y + 6) = 0$ to find $y = -2, -6$ A1\n- Finds $x = 5$ and $x = 11$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Rearranges linear equation for one variable and substitutes into the quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Expands squared bracket and clears fractions correctly.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies to a standard quadratic equation.","code":"M"},{"id":"p4","marks":1,"description":"Finds both correct values for the first variable.","code":"A"},{"id":"p5","marks":1,"description":"Finds both correct values for the second variable.","code":"A"}],"finalAnswers":["x = 5, y = -2; x = 11, y = -6"],"generatedAt":"2026-05-07T02:12:13.942Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"substitution","label":"Substitution","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2x+3y=4$"},{"latex":"$$2x=\\htmlData{anchor=numerator}{\\color{@sky}{4-3y}}$"},{"latex":"$$x=\\htmlData{anchor=xexpr}{\\color{@sky}{\\dfrac{4-3y}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"xexpr"}],"connections":[{"to":"xexpr","from":"numerator","color":"sky","label":"divide by 2"}]},"order":0,"title":"Make x the subject","description":"Rearrange the linear equation so that it can be substituted into the quadratic equation."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@orange}{\\htmlData{anchor=source_x}{x}}=\\dfrac{4-3y}{2}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=replacement}{\\left(\\dfrac{4-3y}{2}\\right)^2}}\\qquad -3y^2=13$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"source_x"},{"color":"orange","style":"text-color","anchor":"replacement"}],"connections":[{"to":"replacement","from":"source_x","color":"orange","label":"substitute for x","style":"solid"}]},"order":1,"title":"Substitute for x","description":"Replace x in the quadratic equation with the expression found from the linear equation. Writing this complete substitution earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\left(\\dfrac{4-3y}{2}\\right)^2-3y^2=13$"},{"latex":"$$\\dfrac{\\htmlData{anchor=expanded}{\\color{@sky}{16-24y+9y^2}}}{4}-3y^2=\\htmlData{anchor=rhs13}{13}$"},{"latex":"$$16-24y+9y^2-12y^2=\\htmlData{anchor=rhs52}{\\color{@sky}{52}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"expanded"},{"color":"sky","style":"text-color-with-background","anchor":"rhs52"}],"connections":[{"to":"rhs52","from":"rhs13","color":"sky","label":"multiply both sides by 4"}]},"order":2,"title":"Expand and clear","description":"Expand the squared bracket, then multiply every term by 4 to clear the denominator. This earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$16-24y+9y^2-12y^2=52$"},{"latex":"$$\\htmlData{anchor=negative}{-3y^2-24y-36}=0$"},{"latex":"$$\\htmlData{anchor=positive}{\\color{@green}{3y^2+24y+36}}=0$"},{"latex":"$$\\htmlData{anchor=simplified}{\\color{@teal}{y^2+8y+12}}=0$"}],"highlights":[{"color":"green","style":"text-color","anchor":"positive"},{"color":"teal","style":"text-color","anchor":"simplified"}],"connections":[{"to":"positive","from":"negative","color":"green","label":"multiply by negative 1"},{"to":"simplified","from":"positive","color":"teal","label":"divide by 3"}]},"order":3,"title":"Form the quadratic","description":"Collect like terms, multiply through by negative 1, then divide by 3. Reaching either of the final two quadratic forms earns the third M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y^2+8y+12=0$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=factors}{(y+2)(y+6)}}=0$"},{"text":"Set each factor equal to zero."},{"latex":"$$y+2=0\\quad\\text{or}\\quad y+6=0$"},{"latex":"$$\\color{@purple}{\\htmlData{anchor=roots}{y=-2\\quad\\text{or}\\quad y=-6}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"factors"},{"color":"purple","style":"text-color","anchor":"roots"}]},"order":4,"title":"Factorise and solve","description":"Find two numbers that multiply to 12 and add to 8. The correct factorisation and both values of y earn A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=yone}{\\color{@teal}{y=-2}}$"},{"latex":"$$x=\\dfrac{4-3\\htmlData{anchor=subone}{\\color{@teal}{(-2)}}}{2}=\\color{@teal}{5}$"},{"latex":"$$\\htmlData{anchor=ytwo}{\\color{@indigo}{y=-6}}$"},{"latex":"$$x=\\dfrac{4-3\\htmlData{anchor=subtwo}{\\color{@indigo}{(-6)}}}{2}=\\color{@indigo}{11}$"},{"latex":"$$\\htmlData{anchor=solutions}{\\color{@green}{(x,y)=(5,-2)\\quad\\text{or}\\quad(11,-6)}}$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"yone"},{"color":"teal","style":"text-color","anchor":"subone"},{"color":"indigo","style":"text-color","anchor":"ytwo"},{"color":"indigo","style":"text-color","anchor":"subtwo"}],"connections":[{"to":"subone","from":"yone","color":"teal","label":"substitute"},{"to":"subtwo","from":"ytwo","color":"indigo","label":"substitute"}]},"order":5,"title":"Find each x-value","description":"Substitute each value of y into the rearranged linear equation. Finding both matching x-values earns the final A1."}]}],"generatedAt":"2026-07-20T16:53:11.064Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__cie","summaries":[{"id":"121b4e20-35c5-4163-bf65-e301615e6d2d","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"d8b1c68f-6bd3-4b56-9555-439c1a785cde","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7d960de9-c383-4ca6-b926-a8586de290be","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"7a998f38-9f5d-49d6-bf36-a741cb70359a","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"57f2dd00-7fa2-491a-b4ff-5a0332e20c8e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"0cc49e8e-d89b-426d-ad24-b2f180fd7bba","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5a91a9ec-0d6f-4980-aa55-d2064da49327","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"05721abb-3610-49f7-aaa6-e71c8d3116a4","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"a16bde18-5d62-42e0-a851-f3075c4dd2f6","difficultyBand":"hard","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true}],"title":"Quadratic Simultaneous Equations","topicId":"9b1881f2-73db-4b0c-bee3-2cb3125ae984","topicSidebar":"$L3c","topicSlug":"grade-8-9-8-1-quadratic-simultaneous-equations","topicTitle":"Quadratic Simultaneous Equations"}]}],"$L3d","$L3e"]}]

Quadratic Simultaneous Equations

Question 3

Quadratic Simultaneous Equations Questions

Quadratic Simultaneous Equations

Question 3

Quadratic Simultaneous Equations Questions