Find the Roots (Zeros) xe^(-x)-2e^(-x)

$x e^{- x} - 2 e^{- x}$

Set $x e^{- x} - 2 e^{- x}$ equal to $0$ .

$x e^{- x} - 2 e^{- x} = 0$

Solve for $x$ .

Factor $e^{- x}$ out of $x e^{- x} - 2 e^{- x}$ .

Factor $e^{- x}$ out of $x e^{- x}$ .

$e^{- x} x - 2 e^{- x} = 0$

Factor $e^{- x}$ out of $- 2 e^{- x}$ .

$e^{- x} x + e^{- x} \cdot - 2 = 0$

Factor $e^{- x}$ out of $e^{- x} x + e^{- x} \cdot - 2$ .

$e^{- x} (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{- x} = 0$

$x - 2 = 0$

Set $e^{- x}$ equal to $0$ and solve for $x$ .

Set $e^{- x}$ equal to $0$ .

$e^{- x} = 0$

Solve $e^{- x} = 0$ for $x$ .

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x}) = \ln (0)$

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

There is no solution for $e^{- x} = 0$

No solution

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $e^{- x} (x - 2) = 0$ true.

$x = 2$