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

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Solution

$3 x + 2 y = 12$
Squaring both sides, we get
${(3 x + 2 y)}^{2} = 144$
$\Rightarrow 9 x^{2} + 4 y^{2} + 2 \times 3 x \times 2 y = 144$
$\Rightarrow 9 x^{2} + 4 y^{2} = 144 - 12 x y$
$\Rightarrow 9 x^{2} + 4 y^{2} = 144 - (12 \times 6)$ since $x y = 6$
$\Rightarrow 9 x^{2} + 4 y^{2} = 144 - 72 = 72$
$∴ 9 x^{2} + 4 y^{2} = 72$

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