If p + q = 4 and p2 + q2 = 10, find p4 + q4 - 2p2q2.
64
Note that \(p^4 + q^4 - 2p^2q^2 = (p^2 - q^2)^2 = \left[(p+q)(p-q)\right]^2\).
From \((p+q)^2 = p^2+q^2+2pq\): \(16 = 10 + 2pq \Rightarrow pq = 3\).
Also \((p-q)^2 = p^2+q^2-2pq = 10 - 6 = 4\), so \((p-q)^2 = 4\).
Thus \((p^2-q^2)^2 = (p+q)^2(p-q)^2 = 16\times4 = 64\).
The value of p4 + q4 - 2p2q2 is 64.
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