Question

If $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $a\ne b$ have one root in common, then their other roots satisfy the equation

• A. $x^2+ax+b=0$
• B. $x^2+bx+ab=0$
• C. $x^2+cx+ab=0$
• D. $x^2+abx+bc=0$

Answer 1

The correct option is C $x^2+cx+ab=0$

$x^2+ax+bc=0\cdots \left(1\right)$
$x^2+bx+ac=0\cdots \left(2\right)$
Subtracting $\left(1\right)$ and $\left(2\right),$ we get
$(a-b)x+c(b-a)=0$
$\Rightarrow x=c$ is common root
$\therefore$ Roots of equation $x^2+ax+bc=0$ are $b,c.$ and
Roots of equation $x^2+bx+ac=0$ are $c,a.$

$\therefore a+b+c=0\Rightarrow a+b=-c$
$\therefore$ Equation with other roots is $x^2-(a+b)x+ab=0$
$\Rightarrow x^2+cx+ab=0$