rra","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"},"tableImages":[],"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","style":"solid"}]},"order":1,"title":"Factor the inequality","description":"Factor the difference of two squares, then keep the inequality sign."},{"type":"text","order":2,"title":"Choose the negative interval","content":"The factors are zero at $p=-4$ and $p=4$. Their product is negative **between** these values: for example, at $p=0$ it is $(0+4)(0-4)=-16$, which is negative.\n\nTherefore\n\n$$\\boxed{-4

Quadratics

Question 10

Quadratics Questions

Quadratics

Question 10

Quadratics Questions