Question

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

• A. $a=b\ne c$
• B. $a\equiv -b=c$
• C. $a=b=c$
• D. None of these

Answer 1

The correct option is C $a=b=c$

Let $x$ & $y$ be roots of the first equation

Let $x,y,z$ be roots of seconds equation

$x+y=-\left(\frac{b}{a}\right)$ from $\left(1\right)$

$x+y+z=-3$ from $\left(2\right)$

substitute $z=\left(\frac{b}{a}\right)-3$ in eq $\left(2\right)$

Solving gives

$a=b$

Thus $x+y=-1......\left(a\right)$

$z=-2$

$xyz=-2\Rightarrow xy=1$

$x=\left(\frac{1}{y}\right)$

Substitute this in $\left(a\right)$

$\therefore a=b=c$