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Question
If x2 + y2 = 29 and xy = 2, find the value of x4 + y4 .
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Solution
We have:
\[\left( x^2 + y^2 \right)^2 = x^4 + 2 x^2 y^2 + y^4 \]
\[ \Rightarrow x^4 + y^4 = \left( x^2 + y^2 \right)^2 - 2 x^2 y^2 \]
\[ \Rightarrow x^4 + y^4 = \left( x^2 + y^2 \right)^2 - 2 \left( xy \right)^2 \]
\[ \Rightarrow x^4 + y^4 = {29}^2 - 2 \left( 2 \right)^2 (\because x^2 + y^2 = 29 \text { and } xy = 2)\]
\[ \Rightarrow x^4 + y^4 = 841 - 8\]
\[ \Rightarrow x^4 + y^4 = 833\]
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