Question

If the roots of the equation $8x^3-14x^2+7x-1=0$ are in G.P., then the roots are

• A. 1, $\frac{1}{2}$, $\frac{1}{4}$
• B. 2, 4, 8
• C. 3, 6, 12
• D. 1,2,-4

Answer 1

The correct option is A

1, $\frac{1}{2}$, $\frac{1}{4}$

Let the roots be $\frac{\alpha }{\beta },\alpha ,\alpha \beta ,\beta \ne 0$
Then the product of roots is ${\alpha }^3=-\frac{-1}{8}=\frac{1}{8}$
⇒ $\alpha =\frac{1}{2}$ and hence β = $\frac{1}{2}$
(Since, sum of roots $=\frac{\alpha }{\beta }+\alpha +\alpha \beta =14/8)$,
so roots are $1,\frac{1}{2}$, $\frac{1}{4}$

Alternate method: By inspecting through options, we get the numbers $1,\frac{1}{2}$, $\frac{1}{4}$ satisfying the given equation.