Precalculus Examples

$r (x) = \frac{x e^{x} + 2 e^{x}}{{(1 + e^{2 x})}^{3}}$

Step 1

Set $\frac{x e^{x} + 2 e^{x}}{{(1 + e^{2 x})}^{3}}$ equal to $0$ .

$\frac{x e^{x} + 2 e^{x}}{{(1 + e^{2 x})}^{3}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Set the numerator equal to zero.

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

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

$e^{x} = 0$

$x + 2 = 0$

Step 2.2.3

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

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Step 2.2.3.1

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

$e^{x} = 0$

Step 2.2.3.2

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

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Step 2.2.3.2.1

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

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

Step 2.2.3.2.2

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

Undefined

Step 2.2.3.2.3

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

No solution

Step 2.2.4

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

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Step 2.2.4.1

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

$x + 2 = 0$

Step 2.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.2.5

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

$x = - 2$

Step 3