Quadratics","calculatorUse":null,"parts":[{"id":"dfb42ecd-d605-446d-9035-1ce2f34a2407","label":"a","content":"By completing the square, find in terms of the constant $p$ the roots of the equation\n\n$$x^2 + px + 4 = 0$$","markscheme":"- $(x + \\frac{p}{2})^2 - \\frac{p^2}{4} + 4 = 0$. M1\n- $(x + \\frac{p}{2})^2 = \\frac{p^2}{4} - 4$. M1\n- $x = -\\frac{p}{2} \\pm \\sqrt{\\frac{p^2}{4} - 4}$ or $x = \\frac{-p \\pm \\sqrt{p^2 - 16}}{2}$. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Completes the square correctly for $x^2 + px$.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges to make $x$ the subject.","code":"M"},{"id":"p3","marks":1,"description":"Final answer $x = \\frac{-p \\pm \\sqrt{p^2 - 16}}{2}$ or equivalent.","code":"A"}],"finalAnswers":["\\frac{-p \\pm \\sqrt{p^2 - 16}}{2}"],"generatedAt":"2026-05-12T02:14:06.561Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"completing_square","label":"Completing the square","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"x^2+px+4=0"},{"latex":"\\left(x+\\frac{p}{2}\\right)^2-\\frac{p^2}{4}+4=0"}]},"order":1,"title":"Complete the square","description":"The coefficient of x is p, so half of it is p over 2. Add and subtract p squared over 4 to keep the equation unchanged; this displayed completed-square form earns the first method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\left(x+\\frac{p}{2}\\right)^2-\\frac{p^2}{4}+4=0"},{"latex":"\\left(x+\\frac{p}{2}\\right)^2=\\frac{p^2}{4}-4"}]},"order":2,"title":"Isolate the square","description":"Move the terms outside the square to the other side. This rearrangement earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\left(x+\\frac{p}{2}\\right)^2=\\frac{p^2}{4}-4"},{"latex":"x+\\frac{p}{2}=\\pm\\sqrt{\\frac{p^2}{4}-4}"}]},"order":3,"title":"Take square roots","description":"Take both square roots. The plus-or-minus sign is essential because either a positive or a negative number can square to the same value."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x+\\frac{p}{2}=\\pm\\sqrt{\\frac{p^2}{4}-4}"},{"latex":"x=-\\frac{p}{2}\\pm\\sqrt{\\frac{p^2}{4}-4}"},{"latex":"x=\\frac{-p\\pm\\sqrt{p^2-16}}{2}"}]},"order":4,"title":"State both roots","description":"Subtract p over 2 from both sides. Including both roots in this final form earns the accuracy mark, A1."}]}],"generatedAt":"2026-07-21T04:07:49.372Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"b5168813-d397-45e9-ad4c-eda9a197808a","label":"b","content":"Hence find the set of values of $p$ for which the equation has no real roots.","markscheme":"- No real roots when $p^2 - 16 < 0$. M1\n- $(p+4)(p-4) < 0 \\Rightarrow -4 < p < 4$. A1\n\n![](https://assets.mathsgenie.co.uk/booklet/87ad5ce57ebbcf67f5d71bbe211f9b3c79f11d6320fcda4872db5b276160cd39.jpg)","marks":2,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"Final answer $-4 < p < 4$.","code":"A"}],"finalAnswers":["-4 < p < 4"],"generatedAt":"2026-05-12T02:14:06.561Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"discriminant_inequality","label":"Solve the discriminant inequality","steps":[{"type":"text","order":0,"title":"Use the discriminant","content":"From the earlier part, the discriminant is $p^2-16$. A quadratic has **no real roots** precisely when its discriminant is negative, so write\n\n$$p^2-16<0.$$\n\nThis is the required condition and earns **M1**.","highlights":[{"text":"Hence find the set of values of $p$ for which the equation has no real roots.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{p^2-16<0}"},{"latex":"\\htmlData{anchor=factored}{\\color{@sky}{(p+4)(p-4)}<0}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"factored"}],"connections":[{"to":"factored","from":"original","color":"sky","label":"factorise"

1.5.3 Quadratic functions

Question 10

1.5.3 Quadratic functions Questions

1.5.3 Quadratic functions

Question 10

1.5.3 Quadratic functions Questions