Find the Roots (Zeros) xe^x+e^x

$x e^{x} + e^{x}$

Step 1

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

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

Step 2

Solve for $x$ .

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

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

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

Multiply by $1$ .

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

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

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

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

$e^{x} = 0$

$x + 1 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = - 1$

Step 3