Question

If the equations $x^2+ax+bc=0$ and $x^2+bx+ca=0$ have a common root and if $a,b$ and $c$ are non-zero distinct real numbers, then their other roots satisfy the equation :

• A. $x^2+x+abc=0$
• B. $x^2-(a+b)x+ab=0$
• C. $x^2+(a+b)x+ab=0$
• D. $x^2+x+ab=0$
• E. $x^2+abx+abc=0$

Answer 1

The correct option is B $x^2-(a+b)x+ab=0$

Given equations are $x^2+ax+bc=0$
and $x^2+bx+ca=0$

On subtracting equation second from equation first, we get
$(a-b)x+c(b-a)=0$
$\Rightarrow (a-b)(x-c)=0$
$\Rightarrow x=c$ is the common root.

Thus, the roots of $x^2+ax+bc=0$ are $b$ and $c$
and that of $x^2+bx+ca=0$ are $c$ and $a.$

It roots are $b$ and $a,$ then equation will be $x^2-(a+b)x+ab=0$.