Question

If the quadratic equations $x^2+bx+ca=0\&x^2+cx+ab=0$ have a common root, the equation containing their other roots is :

• A. $x^2-ax-bc=0$.
• B. $x^2+bx+ac=0$.
• C. $x^2+ax+bc=0$.
• D. $x^2-bx-ac=0$.

Answer 1

Answer: (C)

$x^2+ax+bc=0$.

Let the common root be $\alpha$.
Hence
$\alpha +\beta =-b$
And
$\alpha +\gamma =-c$
Also
$\alpha .\beta =ac$
And
$\alpha .\gamma =ab$
Then
$\frac{\beta }{\gamma }=\frac{c}{b}$
And
$\beta -\gamma =c-b$
Hence
$\beta =c$ and $\gamma =b$.
Therefore the required equation is
$x^2-(\gamma +\beta )x+\gamma .\beta =0$
$x^2-(b+c)x+bc=0$ ...(i)
Now
$\alpha +\beta =-b$
Or
$\alpha +c=-b$
Or
$\alpha =-(b+c)$
and
$\alpha .\beta =ac$
Hence
$\alpha =a$.
Now $\alpha =-(b+c)$
$a=-(b+c)$.
Substituting in the equation, we get
$x^2+ax+bc=0$.