Solve for x e^x+xe^x=0

$e^{x} + x e^{x} = 0$

Step 1

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

Multiply by $1$ .

$e^{x} \cdot 1 + x e^{x} = 0$

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

$e^{x} \cdot 1 + e^{x} x = 0$

Factor $e^{x}$ out of $e^{x} \cdot 1 + e^{x} x$ .

$e^{x} (1 + x) = 0$

Step 2

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

$e^{x} = 0$

$1 + x = 0$

Step 3

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

Step 4

Set $1 + x$ equal to $0$ and solve for $x$ .

Set $1 + x$ equal to $0$ .

$1 + x = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

The final solution is all the values that make $e^{x} (1 + x) = 0$ true.

$x = - 1$