Question

If $x^2+px+r$ and $3x^2+p$ have a common factor, then

• A. $\frac{p^2}{9}+\frac{r}{2}=0$
• B. $\frac{p^3}{27}-\frac{r^2}{4}=0$
• C. $\frac{p^3}{27}+\frac{r^2}{4}=0$
• D. $\frac{p^2}{9}+\frac{r^2}{4}=0$

Answer 1

Answer: (C)

$\frac{p^3}{27}+\frac{r^2}{4}=0$

Let common root be $a$

$⟹a^3+pa+r=0$

$⟹a=\frac{-p\pm \sqrt{p^2-4r}}{2}-\left(1\right)$

Also $3a^2+p⟹a=\sqrt{\frac{-p}{3}}‘-\left(2\right)$

$a^3(a^2+p)+r=0$

$\Rightarrow \sqrt{\frac{-p}{3}}\left[\right(\frac{-p}{3})+p]+r=0$

$\Rightarrow \frac{-p\left(4p^2\right)}{27}=r^2$

$4p^3+27r^2=0$

$\frac{p^3}{27}+\frac{r^2}{4}=0$