If m and n $(m > n)$ are the roots of the equation $7 (x + 2 a)^{2} + 3 a^{2} = 5 a (7 x + 23 a)$ , where $a > 0$ , then what is $3 m - n$ equal to?

12 a

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14 a

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15 a

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18 a

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Solution

The correct option is C $15 a$
$7 (x + 2 a)^{2} + 3 a^{2} = 5 a (7 x + 23 a)$

$7 (x^{2} + 4 a x = 4 a^{2}) + 3 a^{2} = 35 a x + 115 a^{2}$

$7 x^{2} + 28 a x + 28 a^{2} + 3 a^{3} - 35 a x - 115 a^{2} = 0$

$7 x^{2} - 7 a x - 84 a^{2} = 0$

$x^{2} - a x - 12 a^{2} = 0$

$x^{2} - 4 a x + 3 a x - 12 a^{2} = 0$

$x (x - 4 a) + 3 a (x - 4 a) = 0$

$(x - 4 a) (x + 3 a) = 0$

$x = 4 a$ or $x = - 3 a$

as $m > n$

$m = 4 a, n = - 3 a$

Hence, $3 m - n = 3 \times 4 a - (- 3 a) = 15 a$

Option $C$ is correct

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