If ${(6 x + 3 y)}^{2}$ = 576 and xy=6, find the value of $36 x^{2} 9 y^{2}$

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Solution

${(6 x + 3 y)}^{2} = 576$
$x y = 6$
To find:- $36 x^{2} + 9 y^{2} = ?$
Using identity: ${(a + b)}^{2} = a^{2} + b^{2} + 2 a b$
${(6 x + 3 y)}^{2} = {(6 x)}^{2} + {(3 y)}^{2} + 2 (6 x) (3 y)$
$576 = 36 x^{2} + 9 y^{2} + 36 (x y) [∵ {(6 x + 3 y)}^{2} = 576]$
$\Rightarrow 36 x^{2} + 9 y^{2} = 576 - (36 \times 6) [∵ x y = 6]$
$\Rightarrow 36 x^{2} + 9 y^{2} = 576 - 216$
$\Rightarrow 36 x^{2} + 9 y^{2} = 360$
Hence the corect answer is $360$ .

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