Algebra Examples

$x - x e^{5 x + 2} = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Factor $x$ out of $x - x e^{5 x + 2}$ .

Tap for more steps...

Step 1.1.1

Raise $x$ to the power of $1$ .

$x - x e^{5 x + 2} = 0$

Step 1.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x e^{5 x + 2} = 0$

Step 1.1.3

Factor $x$ out of $- x e^{5 x + 2}$ .

$x \cdot 1 + x (- 1 e^{5 x + 2}) = 0$

Step 1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 e^{5 x + 2})$ .

$x (1 - 1 e^{5 x + 2}) = 0$

Step 1.2

Rewrite $- 1 e^{5 x + 2}$ as $- e^{5 x + 2}$ .

$x (1 - e^{5 x + 2}) = 0$

Step 2

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

$x = 0$

$1 - e^{5 x + 2} = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $1 - e^{5 x + 2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $1 - e^{5 x + 2}$ equal to $0$ .

$1 - e^{5 x + 2} = 0$

Step 4.2

Solve $1 - e^{5 x + 2} = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Subtract $1$ from both sides of the equation.

$- e^{5 x + 2} = - 1$

Step 4.2.2

Divide each term in $- e^{5 x + 2} = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $- e^{5 x + 2} = - 1$ by $- 1$ .

$\frac{- e^{5 x + 2}}{- 1} = \frac{- 1}{- 1}$

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{5 x + 2}}{1} = \frac{- 1}{- 1}$

Step 4.2.2.2.2

Divide $e^{5 x + 2}$ by $1$ .

$e^{5 x + 2} = \frac{- 1}{- 1}$

Step 4.2.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.2.3.1

Divide $- 1$ by $- 1$ .

$e^{5 x + 2} = 1$

Step 4.2.3

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

$\ln (e^{5 x + 2}) = \ln (1)$

Step 4.2.4

Expand the left side.

Tap for more steps...

Step 4.2.4.1

Expand $\ln (e^{5 x + 2})$ by moving $5 x + 2$ outside the logarithm.

$(5 x + 2) \ln (e) = \ln (1)$

Step 4.2.4.2

The natural logarithm of $e$ is $1$ .

$(5 x + 2) \cdot 1 = \ln (1)$

Step 4.2.4.3

Multiply $5 x + 2$ by $1$ .

$5 x + 2 = \ln (1)$

Step 4.2.5

Simplify the right side.

Tap for more steps...

Step 4.2.5.1

The natural logarithm of $1$ is $0$ .

$5 x + 2 = 0$

Step 4.2.6

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 4.2.7

Divide each term in $5 x = - 2$ by $5$ and simplify.

Tap for more steps...

Step 4.2.7.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.2.7.2

Simplify the left side.

Tap for more steps...

Step 4.2.7.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 4.2.7.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.2.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 4.2.7.3

Simplify the right side.

Tap for more steps...

Step 4.2.7.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 5

The final solution is all the values that make $x (1 - e^{5 x + 2}) = 0$ true.

$x = 0, - \frac{2}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{2}{5}$

Decimal Form:

$x = 0, - 0.4$