Question

If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Answer 1

$(x-2)$ is a common factor
$\Rightarrow (x=2)$ is common root of both the equation
$\therefore$ putting the value of $x=2$ in both the equation
$\therefore 4+2a+b=0$ and $4+2c+d=0$
On substracting we get
$2(a-c)+b-d=0\Rightarrow 2=-\frac{b-d}{a-c}=\frac{b-d}{c-a}$