If the roots of the cubic equation a{x^3} + b | vedclass

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Question

(a) Let $\frac{A}{R},\;A,\;AR$ be the roots of the equation

$a{x^3} + b{x^2} + cx + d = 0$

then ${A^3} = $Product of the roots $ = - \frac{d}{a}

Vedclass · Accepted Answer

${c^3}a = {b^3}d$

Vedclass · Answer

(a) Let $\frac{A}{R},\;A,\;AR$ be the roots of the equation

$a{x^3} + b{x^2} + cx + d = 0$

then ${A^3} = $Product of the roots $ = - \frac{d}{a}$

$ \Rightarrow $$A = - {\left( {\frac{d}{a}} ight)^{1/3}}$

Since $A$ is a root of the equation.

$	herefore a{A^3} + b{A^2} + cA + d = 0$

$ \Rightarrow $$a\left( { - \frac{d}{a}} ight) + b{\left( { - \frac{d}{a}} ight)^{2/3}} + c{\left( { - \frac{d}{a}} ight)^{1/3}} + d = 0$

==> $b{\left( {\frac{d}{a}} ight)^{2/3}} = c{\left( {\frac{d}{a}} ight)^{1/3}}$             

==> ${b^3}\frac{{{d^2}}}{{{a^2}}} = {c^3}\frac{d}{a}$

==> ${b^3}d = {c^3}a$.

If the roots of the cubic equation $a{x^3} + b{x^2} + cx + d = 0$ are in $G.P.$, then

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