Calculus Examples

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

Step 1

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

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Step 1.1

Multiply by $1$ .

$e^{x} \cdot 1 - 2 x e^{x} = 0$

Step 1.2

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

$e^{x} \cdot 1 + e^{x} (- 2 x) = 0$

Step 1.3

Factor $e^{x}$ out of $e^{x} \cdot 1 + e^{x} (- 2 x)$ .

$e^{x} (1 - 2 x) = 0$

Step 2

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

$e^{x} = 0$

$1 - 2 x = 0$

Step 3

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

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Step 3.1

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

$e^{x} = 0$

Step 3.2

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

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Step 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 3.2.2

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

Undefined

Step 3.2.3

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

No solution

Step 4

Set $1 - 2 x$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $1 - 2 x$ equal to $0$ .

$1 - 2 x = 0$

Step 4.2

Solve $1 - 2 x = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$- 2 x = - 1$

Step 4.2.2

Divide each term in $- 2 x = - 1$ by $- 2$ and simplify.

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Step 4.2.2.1

Divide each term in $- 2 x = - 1$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 1}{- 2}$

Step 4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{2}$

Step 5

The final solution is all the values that make $e^{x} (1 - 2 x) = 0$ true.

$x = \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$