Question

If a,b,c $ϵ$ R and the equations a$x^2+bx+c=0and{x}^3+3x^2+3x+2=0$ have two roots in common, then

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

We have , $x^3+3x^2+3x+2=0$

$\Rightarrow$ $(x+1{)}^3+1=0$

$\Rightarrow$ ( x+1+1) $\left\{\right(x+1{)}^2-(x+1)+1\}=0$

$\Rightarrow$ (x+2) $(x^2+x+1)$ = 0

$\Rightarrow$ x = -2 , $\frac{-1\pm \sqrt{3}i}{2}$ $\Rightarrow$ x = -2 , $\omega$ , ${\omega }^2$ .

Since a,b,c, $ϵ$ R , $a{x}^2+bx+c=0$ cannot have one real and one imaginary root. Therefore, two

common roots of $a{x}^2+bx+c=0$ and $x^3+3x^2+2=0$ are $\omega ,{\omega }^2$ .

Thus, - $\frac{b}{a}=\omega +{\omega }^2=-1$

$\Rightarrow$ a = b and $\frac{c}{a}=\omega ,{\omega }^2=1$ $\Rightarrow$ c = a

$\Rightarrow$ a = b = c