Find the values of $m$ which will make $2 x^{2} + m x y + 3 y^{2} - 5 y - 2$ equivalent to the product of two linear factors

\pm 6

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\pm 7

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\pm 5

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\pm 9

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Solution

The correct option is B $\pm 7$
Given equation is $2 x^{2} + m x y + 3 y^{2} - 5 y - 2$
For it to represent product of two linear factors , it should satisfy the equation $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
Here $a = 2, b = 3, h = \frac{m}{2}, g = 0, f = - \frac{5}{2}, c = - 2$
We have $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = - 12 + 0 - \frac{25}{2} - 0 + \frac{m^{2}}{2} = 0$
$\Rightarrow m^{2} = 49$
$\Rightarrow m = \pm 7$
Therefore correct option is $B$

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