Question

The number of integral value(s) of $x$ satisfying $|x||x-5|=6$ is

Answer 1

$|x||x-5|=6$
$\Rightarrow |x(x-5)|=6$
$\because |a||b|=|ab|$
$\Rightarrow x(x-5)=\pm 6$
$\Rightarrow x^2-5x=\pm 6$

Case 1:
$\Rightarrow x^2-5x=6$
$\Rightarrow x^2-5x-6=0$
$\Rightarrow x^2-6x+x-6=0$
$\Rightarrow (x-6)(x+1)=0$
$\therefore x=-1,6$

Case 2:
$\Rightarrow x^2-5x=-6$
$\Rightarrow x^2-5x+6=0$
$\Rightarrow x^2-2x-3x+6=0$
$\Rightarrow (x-2)(x-3)=0$
$\therefore x=2,3$

Hence, the number of integral solutions of the equation is $4$.