If the roots of the equation $x^3-7 x^2+14 x-8=0$ are in geometric progression, then the difference between the largest and the smallest roots is

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Aicharya · Accepted Answer

Given equation, $x^3-7 x^2+14 x-8=0$ Let the roots of this equation are $\frac{a}{r}, a, a r$. Then, we get Sum of roots $=7$

From Eq. (iii), we get $a=2$ Now, from Eq. ( $i)$, we get $ \begin{aligned} & \frac{2}{r}+2 r=5 \\ & \Rightarrow \quad 2+2 r^2=5 r \\ & \Rightarrow \quad 2 r^2-5 r+2=0 \\ & \Rightarrow \quad 2 r^2-4 r-r+2=0 \\ & \Rightarrow \quad 2 r(r-2)-1(r-2)=0 \\ & \Rightarrow \quad(r-2)(2 r-1)=0 \\ & \Rightarrow \quad r=2 \text { or } r=\frac{1}{2} \end{aligned} $ Thus, the roots are 1, 2, 4 . Hence, difference of largest and smallest number $=4-1=3$

If the roots of the equation $x^3-7 x^2+14 x-8=0$ are in geometric progression, then the difference between…

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