Question

Find the ratio of $b$ to $a$ in order that the equations
$a{x}^2+bx+a=0$ and $x^3-2x^2+2x-1=0$
may have $\left(1\right)$ one, $\left(2\right)$ two roots in common.

Answer 1

$f\left(x\right)=a{x}^2+bx+a=0⟹x^2+\frac{b}{a}x+1=0$

$g\left(x\right)=x^3-2x^2+2x+1=0⟹(x-1)(x^2-x+1)=0$

$g\left(x\right)=0$ has one real root $x=1$ and two imaginary roots

Case 1: For $f\left(x\right)$ and $g\left(x\right)$ to have one common roots, it must be the real root since imaginary roots occurs in conjugate pairs

$\therefore x=1$ is the root of $f\left(x\right)=0$

$\therefore 1+\frac{b}{a}+1=0⟹\frac{b}{a}=-2$

Case 2: For $f\left(x\right)$ and $g\left(x\right)$ to have two common roots, it must be the real root since imaginary roots occurs in conjugate pairs

$x^2+\frac{b}{a}x+1\equiv x^2-x+1⟹\frac{b}{a}=-1$