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 $\ne$ b = c
• C. a = b = c
• D. a = - b = c

Answer 1

The correct option is C a = b = c

$x^3+3x^2+3x+2=0\Rightarrow (x+1{)}^3+1=0\Rightarrow (x+1)=-1,-\omega ,-{\omega }^2\Rightarrow x=-2,-1-\omega ,-1-{\omega }^2\therefore x=-2,{\omega }^2,\omega$
Since given equations have two common roots therefore $\omega$ and ${\omega }^2$ are the common roots of the two equations.
$\therefore$ Equation whose roots are $\omega$ and ${\omega }^2$ is $x^2+x+1=0$
Comparing with $a{x}^2+bx+c=0,$ we get $\frac{a}{1}=\frac{b}{1}=\frac{c}{1}ie.,a=b=c.$