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M1\n- Factorise as $(x + 10)(x - 2) > 0$, identifying the critical values $x=-10$ and $x=2$. M1\n- State $x < -10$ or $x > 2$. A1\n\n![Graph of the parabola y = x^2 + 8x - 20. The parabola opens upwards and intersects the x-axis at x = -10 and x = 2. Arrows on the x-axis point away from the roots, indicating the solution regions x < -10 and x > 2.](https://assets.mathsgenie.co.uk/booklet/e673a7cf46dda185a1c99fdda33cd928e7430eacb119c9411121404bf0874b5c.jpg)","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Rearranges the inequality to $x^2+8x-20>0$.","code":"M"},{"id":"p2","marks":1,"description":"Factorises the quadratic as $(x+10)(x-2)$ and identifies the critical values $-10$ and $2$.","code":"M"},{"id":"p3","marks":1,"description":"States the solution $x<-10$ or $x>2$.","code":"A","acceptable":["$$x\\in(-\\infty,-10)\\cup(2,\\infty)$"]}],"finalAnswers":["$$x < -10$ or $x > 2$"],"generatedAt":"2026-08-31T09:25:35.304Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]},{"id":"28d55e7d-41d1-45c5-bdef-153be433be80","label":"b","content":"Find the set of values for $x$ which satisfy both of the inequalities\n\n$$x^2 + 8x > 20$$\n\n$$18 + 3x < 23 + x$$","markscheme":"- Solve the linear inequality to obtain $x<\\frac{5}{2}$ and combine it with the result from part (a). M1\n- Hence the required set is $x < -10$ or $2 < x < \\frac{5}{2}$. A1","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States the combined solution $x<-10$ or $2M1\n- Factorise as $(x+9)(x-3)<0$. M1\n- Identify the critical values $x=-9$ and $x=3$. A1\n- State $-9A1\n\n![A hand-drawn graph of a parabola opening upwards on a Cartesian coordinate system. The x-axis is horizontal and the y-axis is vertical. The parabola intersects the x-axis at two points, which are labeled with the numbers -9 and 3. The vertex of the parabola is located below the x-axis, between the two intersection points. The parabola is drawn with a double line, emphasizing the region between the roots where the expression is negative.](https://assets.mathsgenie.co.uk/booklet/f2a62e47c1220727d6a0fc57188facf1af9161e2ae9e440cb23b8315fe0b1eb8.jpg)","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Expands and rearranges the inequality to $x^2+6x-27<0$.","code":"M"},{"id":"p2","marks":1,"description":"Factorises the quadratic as $(x+9)(x-3)$.","code":"M"},{"id":"p3","marks":1,"description":"Identifies the critical values $x=-9$ and $x=3$.","code":"A"},{"id":"p4","marks":1,"description":"States the solution $-9M1\n- Simplify to obtain $3x^2-20x+12=0$. A1\n- Factorise and solve to obtain $x=\\frac{2}{3}$ or $x=6$. M1\n- Find the corresponding values of $y$, giving $(x,y)=\\left(\\frac{2}{3},\\frac{7}{3}\\right)$ or $(6,-3)$. A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Substitutes $y=3-x$ into the quadratic equation.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to the quadratic equation $3x^2-20x+12=0$.","code":"A"},{"id":"p3","marks":1,"description":"Solves the quadratic to obtain $x=\\frac{2}{3}$ and $x=6$.","code":"M"},{"id":"p4","marks":1,"description":"Gives both solutions as $\\left(\\frac{2}{3},\\frac{7}{3}\\right)$ and $(6,-3)$.","code":"A"}],"finalAnswers":["$$\\left(\\frac{2}{3},\\frac{7}{3}\\right)$","$$(6,-3)$"],"generatedAt":"2026-08-31T09:25:35.304Z","modelVersion":"cheap:gpt-5.6-sol"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"as_level_maths","summaries":[{"id":"3ae08f67-616b-46a1-b1d0-5ea73b8544f3","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"c150f986-3cc3-4a24-8ac0-8c34c7c47c05","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f23dcedf-2a4c-4de0-847d-e018d9bab020","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a85b0d3d-1b96-4231-8457-8b7746fe4e7c","difficultyBand":null,"calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88b23b6f-8a71-4cdf-94bc-261c4d958f1c","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0255da01-e68c-4e51-b514-545fa8f8f9ce","difficultyBand":null,"calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"26cac68e-94a4-4d2d-8895-1baed6671fea","difficultyBand":null,"calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"be9975e4-05a7-44bf-97cf-d78a6b2efa9d","difficultyBand":null,"calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true}],"title":"3.5 Quadratic Inequalities","topicId":"02913333-61fa-49da-a356-ee3dfb3af099","topicSidebar":"$L3c","topicSlug":"as-pure-mathematics-3-5-quadratic-inequalities","topicTitle":"3.5 Quadratic Inequalities"}]}],"$L3d","$L3e"]}]