Question

If the equation $a{x}^2+bx+c=0$ and $x^3+3x^2+3x+2=0$ have two common roots, then

• A. $a=b\ne c$
• B. $a=-b=c$
• C. $a=b=c$
• D. $a+b+c=3$

Answer 1

Answer: (C), (D)

The correct options are
C $a=b=c$
D $a+b+c=3$

$x^3+3x^2+3x+2=0$
$(x+1{)}^3+1=0$
$(x+1+1)\left(\right(x+1{)}^2+1-(x+1))=0$
$(x+2)(x^2+2x+1+1-x-1)=0$
$(x+2)(x^2+x+1)=0$
Hence,
$x=-2,w,w^2$
Hence, the common roots are $w,w^2$ since either one of them cannot occur by itself. They are conjugate pairs.
Substituting in 1, we get
$a=b=c=1$
Hence,
$a+b+c=3\left(1\right)=3$.