Calculus Examples

$4 e^{x} x + 8 e^{x} = 0$

Step 1

Reorder factors in $4 e^{x} x + 8 e^{x}$ .

$4 x e^{x} + 8 e^{x} = 0$

Step 2

Factor $4 e^{x}$ out of $4 x e^{x} + 8 e^{x}$ .

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Step 2.1

Factor $4 e^{x}$ out of $4 x e^{x}$ .

$4 e^{x} (x) + 8 e^{x} = 0$

Step 2.2

Factor $4 e^{x}$ out of $8 e^{x}$ .

$4 e^{x} (x) + 4 e^{x} (2) = 0$

Step 2.3

Factor $4 e^{x}$ out of $4 e^{x} (x) + 4 e^{x} (2)$ .

$4 e^{x} (x + 2) = 0$

Step 3

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

$e^{x} = 0$

$x + 2 = 0$

Step 4

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

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Step 4.1

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

$e^{x} = 0$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

Undefined

Step 4.2.3

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

No solution

Step 5

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

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Step 5.1

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

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

The final solution is all the values that make $4 e^{x} (x + 2) = 0$ true.

$x = - 2$