Find the range of b

Posted January 17th, 2010 by alpha

If the roots of then equation $x^3+bx^2+cx-1=0$ form an increasing G.P. , then find the range of 'b'?

Comments

On January 17th, 2010 Structure says:

Let be $q>0$ the ratio and

$x_1=\frac{u}{q}$

$x_2=u$

$x_3=qu$

Viete relations are

$x_1+x_2+x_3=-b$

$x_1x_2x_3=1$

$\frac{u}{q}+u+uq=-b$

$\frac{u}{q}*u* uq=1$

$u^3=1$

has a single real root $u=1$

$-b=\frac{1}{q}+1+q\ge 3$

$b\in (-\infty,-3]$

I have used

$q+\frac{1}{q}\ge 2$