Question

If roots of $x^3+b{x}^2+cx+d=0$ are-

• A. In A.,P. then $2b^3-9bc+27d=0$
• B. In G.P. then $b^{3}d=c^3$
• C. In G.P. then $27d^3=9bc{d}^2-4c^{3}d$
• D. Equal then $c^3=b^3+3bc$

Answer 1

Answer: (A), (B)

In G.P. then $b^{3}d=c^3$
In A.,P. then $2b^3-9bc+27d=0$

Let roots be $\alpha -\beta ,\alpha ,\alpha +\beta$, so that

$3\alpha =-b\Rightarrow \alpha =-b/3$

$\therefore -\frac{b^3}{27}+\frac{b^3}{9}-\frac{bc}{3}+d=0$

$\Rightarrow 2b^3-9bc+27d=0$

Next, roots be $\alpha /\beta ,\alpha ,\alpha \beta$, so that

${\alpha }^3=-d$ or $\alpha =(-d{)}^{1/3}$

$\therefore -d+b(-d{)}^{2/3}+c(-d{)}^{1/3}+d=0$

$\Rightarrow b^{3}d=c^3$