Pre-Algebra Examples

$2 t e^{- 7 t} - 7 t^{2} e^{- 7 t} = 0$

Step 1

Factor $t e^{- 7 t}$ out of $2 t e^{- 7 t} - 7 t^{2} e^{- 7 t}$ .

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Step 1.1

Factor $t e^{- 7 t}$ out of $2 t e^{- 7 t}$ .

$t e^{- 7 t} (2) - 7 t^{2} e^{- 7 t} = 0$

Step 1.2

Factor $t e^{- 7 t}$ out of $- 7 t^{2} e^{- 7 t}$ .

$t e^{- 7 t} (2) + t e^{- 7 t} (- 7 t) = 0$

Step 1.3

Factor $t e^{- 7 t}$ out of $t e^{- 7 t} (2) + t e^{- 7 t} (- 7 t)$ .

$t e^{- 7 t} (2 - 7 t) = 0$

Step 2

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

$t = 0$

$e^{- 7 t} = 0$

$2 - 7 t = 0$

Step 3

Set $t$ equal to $0$ .

$t = 0$

Step 4

Set $e^{- 7 t}$ equal to $0$ and solve for $t$ .

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Step 4.1

Set $e^{- 7 t}$ equal to $0$ .

$e^{- 7 t} = 0$

Step 4.2

Solve $e^{- 7 t} = 0$ for $t$ .

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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.

$\ln (e^{- 7 t}) = \ln (0)$

Step 4.2.2

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

Undefined

Step 4.2.3

There is no solution for $e^{- 7 t} = 0$

No solution

Step 5

Set $2 - 7 t$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $2 - 7 t$ equal to $0$ .

$2 - 7 t = 0$

Step 5.2

Solve $2 - 7 t = 0$ for $t$ .

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Step 5.2.1

Subtract $2$ from both sides of the equation.

$- 7 t = - 2$

Step 5.2.2

Divide each term in $- 7 t = - 2$ by $- 7$ and simplify.

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Step 5.2.2.1

Divide each term in $- 7 t = - 2$ by $- 7$ .

$\frac{- 7 t}{- 7} = \frac{- 2}{- 7}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{- 7 t}{- 7} = \frac{- 2}{- 7}$

Step 5.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 2}{- 7}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

$t = \frac{2}{7}$

Step 6

The final solution is all the values that make $t e^{- 7 t} (2 - 7 t) = 0$ true.

$t = 0, \frac{2}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = 0, \frac{2}{7}$

Decimal Form:

$t = 0, 0.\overline{285714}$