Signum Function

Algebra Level 3

The signum function is defined to be $\text{sgn}(x)=1,0,-1$ if $x>0$ , $x=0$ and $x<0$ respectively. Find the number of solutions of the equation $x(x-3\text{sgn}(x))+\text{sgn}^2(x)=0$

The answer is 5.

2 solutions

Renz Mina
Jul 31, 2016

Use quadratic formula to solve for $x$ in terms of $sgn(x)$ . We get that $x=\frac{3\pm \sqrt{5}}{2}sgn(x)$ . We get $x=\pm\frac{3\pm\sqrt{5}}{2}$ or $x=0$ . So there are $5$ solutions.

How come you can write x*sgn(x)=x?

Renz Mina - 4 years, 10 months ago

Got it. It should have been x*sgn(x) =|x|

Mayank Jain - 4 years, 10 months ago

What if we multiply all the terms and then write x*sgn(x)=x and take two cases where in [sgn(x)]^2 is either 0 or 1?

Mayank Jain - 4 years, 10 months ago

As Renz hinted, it is not true that " x * sgn (x) = x ".

So, "What if a false statement is assumed to be true?" Then, anything can be true.

Calvin Lin Staff - 4 years, 10 months ago

Hải Trung Lê
Aug 5, 2016

Just consider 3 cases, $x>0,\ x<0,\ x=0$ and then solve the quadratic equations. The first two cases give us $4$ solutions, and $x=0$ also a valid solution. So there are $\boxed{5}$ solutions.

Signum Function

The answer is 5.

2 solutions

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