The following expression $36 x^{2} + 96 x y + 64 y^{2}$ has a solution is $(k x + 8 y)^{2}$ . Find the value of k.

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Solution

$36 x^{2} + 96 x y + 64 y^{2}$

$= 36 x^{2} + 64 y^{2} + 96 x y$

$= {(6 x)}^{2} + {(8 y)}^{2} + 2 \times 6 x \times 8 x - 2 \times 6 x \times 8 x + 96 x y$

$= {(6 x + 8 y)}^{2} - 96 x y + 96 x y$

$= {(6 x + 8 y)}^{2}$

Comparing with $(k x + 8 y)^{2}$ , we get $k = 6$

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