If 3x + 2y = 12 and xy = 6, find the value of $9 x^{2} + 4 y^{2}$

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Solution

We have,
${(3 x + 2 y)}^{2} = {(3 x)}^{2} + {(2 y)}^{2} + 2 \times 3 x \times 2 y$
$\Rightarrow {(3 x + 2 y)}^{2} = 9 x^{2} + 4 y^{2} + 12 x y$

$\Rightarrow 12^{2} = 9 x^{2} + 4 y^{2} + 12 \times 6$ [Putting 3x + 2y = 12 and xy = 6]

$\Rightarrow 144 = 9 x^{2} + 4 y^{2} + 72$

$\Rightarrow 144 - 72 = 9 x^{2} + 4 y^{2}$

$\Rightarrow 9 x^{2} + 4 y^{2} = 72$

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