If the equation $a x^{3} + 3 b x^{2} + 3 c x + d = 0$ has two equal roots, show that each of them is equal to $\frac{b c - a d}{2 (a c - b^{2})}$ .

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Solution

By the conditions of the question,
$a x^{3} + 3 b x^{2} + 3 c x + d = 0$ and its first derived equation $a x^{2} + 2 b x + c = 0$ must have a common root.
Hence $b x^{2} + 2 c x + d = 0$ and $a x^{2} + 2 b x + c = 0$ must have a common root
$\frac{x^{2}}{2 c^{2} - 2 b d} = \frac{x}{a d - b c} = \frac{1}{2 b^{2} - 2 a c}$
$4 (c^{2} - b c) (b^{2} - a c) = {(a d - b c)}^{2}$
$x = \frac{a d - b c}{2 (b^{2} - a c)}$
$= \frac{b c - a d}{2 (a c - b^{2})}$

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