difficultyBand":"medium","totalMarks":5,"sourceTitle":"Quadratic Simultaneous Equations","calculatorUse":null,"isCalculation":true,"parts":[{"id":"10220e66-fc92-4f57-a4f9-d8ddf1d34922","label":"0","content":"Solve the simultaneous equations\n\n$$x^2 + y^2 = 17$$\n\n$$y = x - 3$$","markscheme":"- Substitutes $y = x - 3$ into the quadratic: $x^2 + (x - 3)^2 = 17$ M1\n- Simplifies to $2x^2 - 6x - 8 = 0$ or $x^2 - 3x - 4 = 0$ M1\n- Factorises to $(x - 4)(x + 1) = 0$ M1\n- Finds $x = 4$ and $x = -1$ A1\n- Finds $y = 1$ and $y = -4$ A1","marks":5,"order":0,"markingRubric":{"totalMarks":5,"points":[{"id":"p1","marks":1,"description":"Substitutes the linear expression for $y$ into the quadratic equation.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the resulting equation into a quadratic equal to zero.","code":"M"},{"id":"p3","marks":1,"description":"Correctly factorises or uses the quadratic formula.","code":"M"},{"id":"p4","marks":1,"description":"Finds both correct values for $x$.","code":"A"},{"id":"p5","marks":1,"description":"Finds both correct corresponding values for $y$.","code":"A"}],"finalAnswers":["x = 4, y = 1; x = -1, y = -4"],"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":"17=x^2+\\htmlData{anchor=source_y_squared}{\\color{@sky}{y^2}}"},{"latex":"17=x^2+\\htmlData{anchor=replaced_square}{\\color{@sky}{(x-3)^2}}"},{"latex":"x^2+(x-3)^2=17"}],"connections":[{"to":"replaced_square","from":"source_y_squared","color":"sky","label":"replace y","style":"solid"}]},"order":0,"title":"Substitute for y","description":"Since y = x - 3, replace y in the first equation. This substitution earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x^2+x^2-6x+9=\\htmlData{anchor=expanded}{17}"},{"latex":"\\htmlData{anchor=collected}{2x^2-6x-8=0}"},{"latex":"\\htmlData{anchor=divided}{x^2-3x-4=0}"}],"connections":[{"to":"collected","from":"expanded","color":"orange","label":"collect terms and subtract 17","style":"solid"},{"to":"divided","from":"collected","color":"teal","label":"divide by 2","style":"solid"}]},"order":1,"title":"Simplify the quadratic","description":"Expand the bracket, collect like terms, then make the right-hand side zero. This earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x^2-3x-4=\\htmlData{anchor=quadratic}{0}"},{"latex":"\\htmlData{anchor=factorised}{(x-4)(x+1)=0}"}],"connections":[{"to":"factorised","from":"quadratic","color":"purple","label":"factorise","style":"solid"}]},"order":2,"title":"Factorise the quadratic","description":"Find two numbers which multiply to negative four and add to negative three: negative four and one. This factorisation earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=factors}{(x-4)(x+1)=0}"},{"latex":"\\htmlData{anchor=solutions}{x=4\\quad\\text{or}\\quad x=-1}"}],"connections":[{"to":"solutions","from":"factors","color":"green","label":"set each factor to 0","style":"solid"}]},"order":3,"title":"Solve for x","description":"Set each factor equal to zero. Finding both x-values earns A1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=rule}{y=x-3}"},{"latex":"\\htmlData{anchor=first}{y=4-3=1}"},{"latex":"\\htmlData{anchor=second}{y=-1-3=-4}"}],"connections":[{"to":"first","from":"rule","color":"sky","label":"use x = 4","style":"solid"},{"to":"second","from":"rule","color":"orange","label":"use x = -1","style":"solid"}]},"order":4,"title":"Find the y-values","description":"Substitute each x-value into y = x - 3. Giving the matching two y-values earns the final A1. So the solutions are (x, y) = (4, 1) and (-1, -4)."}]}],"generatedAt":"2026-07-21T04:03:35.670Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":54},"walkthr

Quadratic Simultaneous Equations

Question 1

Quadratic Simultaneous Equations Questions

Quadratic Simultaneous Equations

Question 1

Quadratic Simultaneous Equations Questions