Question 42: Simplify the expression: \( C = 9{x^2} – 2xy + \frac{1}{9}{y^2} – 2\left( {3x – \frac{1}{3} y} \right)\left( {3x + \frac{1}{3}y} \right) + {\left( {3x + \frac{1}{3}y} \right)^2}\)

\(\begin{array}{l} C = 9{x^2} – 2xy + \frac{1}{9}{y^2} – 2\left( {3x – \frac{1}{3} y} \right)\left( {3x + \frac{1}{3}y} \right) + {\left( {3x + \frac{1}{3}y} \right)^2}\\ \Leftrightarrow C = {(3x)^2} – 2.3x.\frac{1}{3}y + {\left( {\frac{1}{3}y} \right)^2} – 2\left ( {3x + \frac{1}{3}y} \right)\left( {3x – \frac{1}{3}y} \right) + {\left( {3x + \frac{1}{ 3}y} \right)^2}\\ \Leftrightarrow C = {\left( {3x – \frac{1}{3}y} \right)^2} – 2\left( {3x + \frac{ 1}{3}y} \right)\left( {3x – \frac{1}{3}y} \right) + {\left( {3x + \frac{1}{3}y} \right) ^2}\\ \Leftrightarrow C = {\left( {\left( {3x – \frac{1}{3}y} \right) – \left( {3x + \frac{1}{3}y} \right)} \right)^2}\\ \Leftrightarrow C = {\left( {3x – \frac{1}{3}y – 3x – \frac{1}{3}y} \right)^2 }\\ \Rightarrow C = {\left( { – \frac{2}{3}y} \right)^2} = \frac{4}{9}{y^2} \end{array}\)