Question

Using a suitable identity, find the following products
[2x/3y + 3x/2y] [2x/3y + 3x/2y]​

Answer 1

$\color{red} {\mathfrak{ \huge \underline{Question}}}$

$\rm \: \to \: using \: a \: suitable \: identity \: find \: the \: product$

$\rm \{ \frac{2x}{3y} + \frac{3x}{2y} \} \{ \frac{2x}{3y} + \frac{3x}{2y} \}$

$\color{red} {\mathfrak{ \huge \underline{solution}}}$

$\rm \{ \frac{2x}{3y} + \frac{3x}{2y} \} \{ \frac{2x}{3y} + \frac{3x}{2y} \}$

$\rm \{ \frac{2x}{3y} + \frac{3x}{2y} \} {}^{2}$

$\rm\: by \: using \: this \: identity$

$\implies\boxed{\green{ \rm \: (a + b) {}^{2} = {a}^{2} + {b}^{2} + 2ab }}$

$\to \rm \: ( \frac{2x}{3y} ) {}^{2} + ( \frac{3x}{2y} ) {}^{2} + 2 \times \frac{2x}{3y} \times \frac{3x}{2y}$

$\rm \: \to \: \frac{4 {x}^{2} }{9 {y}^{2} } + \frac{9 {x}^{2} }{4 {y}^{2} } + 2 \times \frac{6 {x}^{2} }{6 {y}^{2} }$

$\rm \: \to \: \frac{4 {x}^{2} }{9 {y}^{2} } + \frac{9 {x}^{2} }{4 {y}^{2} } + \frac{2 {x}^{2} }{ {y}^{2} }$

$\rm \: taking \: common \: \: \: \frac{ {x}^{2} }{ {y}^{2} }$

$\to \: \rm \: \frac{x {}^{2} }{ {y}^{2} } ( \frac{4}{9} + \frac{9}{4} + \frac{2}{1} )$

$\rm \: taking \: lcm \: is \: 36$

$\rm\to \frac{x {}^{2} }{ {y}^{2} } ( \frac{4 \times 4 + 9 \times 9 + 2 \times 36}{36} )$

$\rm\to \frac{x {}^{2} }{ {y}^{2} }( \frac{16 + 81 + 72}{36} )$

$\rm\to \frac{x {}^{2} }{ {y}^{2} }( \frac{169}{36} )$

$\rm\to \frac{169x {}^{2} }{ 36{y}^{2} }$

$\color{red} {\mathfrak{ \huge \underline{answer}}}$

$\implies\boxed{\blue{ \rm \: {\frac{169x {}^{2} }{ 36{y}^{2} }}}}$

Answer 2

$\sf{\huge{\bold{\pink{\bigstar{\boxed{\boxed{Solution}}}}}}}$

$\sf\implies [\dfrac{2x}{3y} + \dfrac{3x}{2y}] [\dfrac{2x}{3y} + \dfrac{3x}{2y}]$

$\sf\implies [\dfrac{2x}{3y} + \dfrac{3x}{2y}]^2$

$\sf{\red{\boxed{\bold{( a + b )^2= a^2+ b^2 + 2ab }}}}$

$\sf\implies [(\dfrac{2x}{3y})^2 + (\dfrac{3x}{2y})^2 + 2\times \dfrac{2x}{3y}\times \dfrac{3x}{2y}]$

$\sf\implies [\dfrac{4x^2}{9y^2} + \dfrac{9x^2}{4y^2} +\dfrac{ 2x^2}{y^2}]$

$\sf\implies\dfrac{ 4\times 4x^2 + 9\times 9x^2+ 2\times 36x^2 }{ 36 y^2}$

$\sf\implies\dfrac{169x^2}{36y^2}$

$\sf{\blue{\boxed{\bold{\dag \dfrac{169x^2}{36y^2} }}}}$