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Given p + q and p^4 + q^4: find pq and the quadratic or expression it gives

The same question, asked 2 times: 2021 Paper 1*, 26 Jun 2022. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 1

    78
    Let pp and qq be two positive number such that p+q=2p+q= 2 and p4+q4=272p^{4}+q^{4}= 272. Then pp and qq are roots of the equation :

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  2. Question 2

    26 Jun 2022

    172
    Let pp and qq be two real numbers such that p+q=p + q = 3 and p4+q4=369p^{4}+q^{4}= 369. Then (1p+1q)−2\left( \frac{1}{p}+\frac{1}{q} \right)^{- 2} is equal to