Question

If x² + y² = 29 and xy = 2, find the value of x⁴ + y⁴ .

Answer 1

We have:

\[\left( x^2 + y^2 \right)^2 = x^4 + 2 x^2 y^2 + y^4 \]

\[ \Rightarrow x^4 + y^4 = \left( x^2 + y^2 \right)^2 - 2 x^2 y^2 \]

\[ \Rightarrow x^4 + y^4 = \left( x^2 + y^2 \right)^2 - 2 \left( xy \right)^2 \]

\[ \Rightarrow x^4 + y^4 = {29}^2 - 2 \left( 2 \right)^2 (\because x^2 + y^2 = 29 \text { and } xy = 2)\]

\[ \Rightarrow x^4 + y^4 = 841 - 8\]

\[ \Rightarrow x^4 + y^4 = 833\]