Which of the following is/are the roots of $(x - 1) (x - 2) (3 x - 1) (3 x - 2) = 8 x^{2}$ ?

\frac{9 - \sqrt{57}}{6}

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\frac{9 - \sqrt{57}}{3}

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\frac{9 + \sqrt{57}}{6}

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\frac{9 + \sqrt{57}}{3}

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Solution

The correct option is C $\frac{9 + \sqrt{57}}{6}$
The given equation is $(x - 1) (x - 2) (3 x - 1) (3 x - 2) = 8 x^{2}$ .
This equation can be rearranged to as:
$(x - 1) (3 x - 2) (x - 2) (3 x - 1) = 8 x^{2},$
$\Rightarrow (3 x^{2} - 5 x + 2) (3 x^{2} - 7 x + 2) = 8 x^{2}$
Dividing the equation by $x^{2},$ we get:
$\Rightarrow (3 x - 5 + \frac{2}{x}) (3 x - 7 + \frac{2}{x}) = 8$
Assuming $3 x + \frac{2}{x} = t$
$\Rightarrow (t - 5) (t - 7) = 8$
$t^{2} - 12 t + 35 = 8$
$\Rightarrow t^{2} - 12 t + 27 = 0$
$\Rightarrow (t - 9) (t - 3) = 0$
$\Rightarrow t = 3, 9$
Case 1: $3 x + \frac{2}{x} = 3$
$\Rightarrow 3 x^{2} + 2 - 3 x = 0$
$\Rightarrow 3 x^{2} - 3 x + 2 = 0$
$D = (- 3)^{2} - 4 (3) (2)$
$D = 9 - 24 < 0 \Rightarrow$ No real roots

Case 2: $3 x + \frac{2}{x} = 3$
$\Rightarrow 3 x^{2} - 9 x + 2 = 0$
$D = (- 9)^{2} - 4 (3) (2)$
$= 81 - 24 = 57 \Rightarrow$ Real and distinct roots
$\Rightarrow x = \frac{9 \pm \sqrt{57}}{6}$ are the roots.

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