Question

Find the value of the following expression:  81x² + 16y² − 72xy, when \[x = \frac{2}{3}\] and  \[y = \frac{3}{4}\]

Answer 1

Let us consider the following expression: \[81 x^2 + 16 y^2 - 72xy\]

Now

\[81 x^2 + 16 y^2 - 72xy = \left( 9x - 4y \right)^2\]         (Using identity \[\left( a + b \right)^2 = a^2 - 2ab + b^2\])

\[\Rightarrow 81 x^2 + 16 y^2 - 72xy = \left[ 9\left( \frac{2}{3} \right) - 4\left( \frac{3}{4} \right) \right]^2 (\text { Substituting } x = \frac{2}{3}\text { and y } = \frac{3}{4})\]

\[ \Rightarrow 81 x^2 + 16 y^2 - 72xy = \left[ 6 - 3 \right]^2 \]

\[ \Rightarrow 81 x^2 + 16 y^2 - 72xy = 3^2 \]

\[ \Rightarrow 81 x^2 + 16 y^2 - 72xy = 9\]