Question

Find number of solutions of the equation :
$sgn( {x}^{2} - 3x + 2) = 2x - 1$
where sgn (•) is signum function of •​

Answer 1

We are given an equation:

$sgn(x^{2} -3x+2) = 2x-1$

First of all, let us have a look at the solutions of signum function.

Signum function returns the sign (0, positive or negative), depending on the input given.

$sgn(x^{2} -3x+2)$ has a quadratic equation, $x^{2} -3x+2$.

Solving this quadratic equation:

$x^{2} -2x-x+2\\x(x-2) -1(x-2)\\(x-1)(x-2)$

So, 3 values are possible:

$sgn(x^{2} -3x+2)$ = 0 for x = 1 and x = 2,

$sgn(x^{2} -3x+2)$ = -1 for {1 < x < 2}

$sgn(x^{2} -3x+2)$ = 1 for x<1 or x>2

So, putting the values of signum function in given equation and finding out solutions.

$sgn(x^{2} -3x+2) = 2x-1$

1. Putting values of signum function = 0

$0=2x-1\\\Rightarrow 2x=1\\\Rightarrow x=\dfrac{1}{2}$

2. Putting values of signum function = 1

$1 =2x-1\\\Rightarrow 2x=2\\\Rightarrow x=1$

3. Putting values of signum function = -1

$-1 =2x-1\\\Rightarrow 2x=0\\\Rightarrow x=0$

So, the solutions for $x$ using the given equation are:

$x=0, \dfrac{1}{2}, 1$