Question

The number of non-zero solutions of the equation $x^2-5x-6sgn\left(x\right)=0$ is
Note: $sgn\left(x\right)$ denotes the signum function.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

$sgn\left(x\right)=\{\begin{array}{c}-1,x<0\\ 0,x=0\\ 1,x>0\end{array}$
Case I: When $x<0$, we get the equation as $x^2-5x+6=0$.
The roots of this equation are $2$ and $3$.
But, they don't satisfy the initial condition: $x<0$.

So, no solution when $x<0$.

Case II: When $x>0$ we get the equation as $x^2-5x-6=0$.
The roots of this equation are $6$ and $-1$.
Only $6$ satisfies the initial condition: $x>0$.
So, in total we get one solution of the given equation.

And $x\ne 0$ as it is given in the question.

Hence, option A.