If the roots of $x^3-42 x^2+336 x-512=0$, are in increasing geometric progression, then its common ratio is

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Aicharya · Accepted Answer

Given, cubic equation is $ \begin{aligned} & x^3-42 x^2+336 x-512=0 \ & \Rightarrow x^2(x-2)-40 x(x-2)+256(x-2)=0 \ & \Rightarrow \quad(x-2)\left(x^2-40 x+256ight)=0 \ & \end{aligned} $ $ \begin{array}{rlrl} \Rightarrow & (x-2)\left\{x^2-32 x-8 x+256ight\} & =0 \ \Rightarrow & & (x-2)\{x(x-32)-8(x-32)\} & =0 \ \Rightarrow & & (x-2)(x-32)(x-8) & =0 \ \Rightarrow & & (x-2)(x-8)(x-32) & =0 \ \Rightarrow & & x=2,8,32 \end{array} $ Which represents a geometric progression in increasing order. Common ratio $=\frac{8}{2}=4: 1$

If the roots of $x^3-42 x^2+336 x-512=0$, are in increasing geometric progression, then its common ratio is

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