If $2 x + y = 14$ and $x y = 6$ , Find the value of $4 x^{2} + y^{2}$ .

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Solution

Given $(i) 2 x + y = 14$
and $(i i) x y = 6$

Using $(i)$
$(2 x + y)^{2} = 14^{2}$
Since $(a + b)^{2} = a^{2} + b^{2} + 2 a b$
$⟹ (2 x + y)^{2}$ = $(2 x)^{2} + y^{2} + (2 \times 2 x \times y) = 14^{2} = 196$
$⟹ 4 x^{2} + y^{2} + 4 x y = 196$
Using $(i i)$
$⟹ 4 x^{2} + y^{2} + 4 \times 6 = 196$
$⟹ 4 x^{2} + y^{2} + 24 = 196$
$⟹ 4 x^{2} + y^{2} = 196 - 24$
$⟹ 4 x^{2} + y^{2} = 172$
Hence, $4 x^{2} + y^{2} = 172$

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