Question

The number of non-zero solution(s) of the equation $x^2-5x-6\text{sgn}\left(x\right)=0$ is (where $\text{sgn}$ represents the signum function)

• A. $2$
• B. $3$
• C. $1$
• D. $0$

Answer 1

The correct option is C $1$

$x^2-5x-6\text{sgn}\left(x\right)=0$
We know that,
$\text{sgn}\left(x\right)=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$

Case $1:$ $x<0$
$x^2-5x+6=0$
$\Rightarrow (x-2)(x-3)=0$
$\Rightarrow x=2,3$
But $x<0$
So, there is no solution.

Case $2:$ $x=0$
$x^2-5x=0$
$\Rightarrow x(x-5)=0$
$\Rightarrow x=0[x=5\text{rejected}]$

Case $3:$ $x>0$
$x^2-5x-6=0$
$\Rightarrow (x+1)(x-6)=0$
$\Rightarrow x=6[x=-1\text{rejected}]$

Hence, the solutions of the given equation is $0,6$
The only non-zero solution is $6.$