Question

If 3x + 5y = 11 and xy = 2, find the value of 9x² + 25y²

Answer 1

We have:

\[\left( 3x + 5y \right)^2 = \left( 3x \right)^2 + 2\left( 3x \right)\left( 5y \right) + \left( 5y \right)^2 \]

\[ \Rightarrow \left( 3x + 5y \right)^2 = 9 x^2 + 30xy + 25 y^2 \]

\[ \Rightarrow 9 x^2 + 25 y^2 = \left( 3x + 5y \right)^2 - 30xy\]

\[\Rightarrow 9 x^2 + 25 y^2 = {11}^2 - 30 \times 2\]     (\[\because\] \[3x + 5y = 11 \text { and } xy = 2\])

\[\Rightarrow 9 x^2 + 25 y^2 = 121 - 60\]

\[ \Rightarrow 9 x^2 + 25 y^2 = 61\]