If the equation $x^{4} + {ax}^{3} + {bx}^{2} + cx + d = 0$ has three equal roots , then that root is

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$\frac{6 c - ab}{3 a^{2} - 4 b}$

$\frac{6 c - ab}{8 b - 3 a^{2}}$

$\frac{6 c - ab}{3 a^{2} - 8 b}$

$\frac{ab - 6 c}{8 b + 3 a^{2}}$

answer is D.

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Detailed Solution

$x^{4} + {ax}^{3} + {bx}^{2} + cx + d = 0$ has three equal roots then

$S_{1} = 3 α + β = - a; S_{2} = 3 α (α + β) = b; S_{3} = α^{2} (α + 3 β) = - c; S_{4} = α^{3} β$

$\begin{aligned} 6 c - a b = α (3 α^{2} - 6 α β + 3 β^{2}) \\ 3 a^{2} - 8 b = 3 α^{2} - 6 α β + 3 β^{2} \end{aligned}$

$∴ \frac{6 c - a b}{3 a^{2} - 8 b} = \frac{α (3 α^{2} - 6 a β + 3 β^{2})}{3 α^{2} - 6 α β + 3 β^{2}} = α$

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