Algebra Examples

$- 3 x^{2} e^{- 2 x} + 3 x e^{- 2 x}$

Step 1

Set $- 3 x^{2} e^{- 2 x} + 3 x e^{- 2 x}$ equal to $0$ .

$- 3 x^{2} e^{- 2 x} + 3 x e^{- 2 x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $3 x e^{- 2 x}$ out of $- 3 x^{2} e^{- 2 x} + 3 x e^{- 2 x}$ .

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Step 2.1.1

Factor $3 x e^{- 2 x}$ out of $- 3 x^{2} e^{- 2 x}$ .

$3 x e^{- 2 x} (- x) + 3 x e^{- 2 x} = 0$

Step 2.1.2

Factor $3 x e^{- 2 x}$ out of $3 x e^{- 2 x}$ .

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

Step 2.1.3

Factor $3 x e^{- 2 x}$ out of $3 x e^{- 2 x} (- x) + 3 x e^{- 2 x} (1)$ .

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

Step 2.2

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

$x = 0$

$e^{- 2 x} = 0$

$- x + 1 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

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

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Step 2.4.1

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

$e^{- 2 x} = 0$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

Undefined

Step 2.4.2.3

There is no solution for $e^{- 2 x} = 0$

No solution

Step 2.5

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

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Step 2.5.1

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

$- x + 1 = 0$

Step 2.5.2

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

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Step 2.5.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 2.5.2.2

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

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.2.2.2

Divide $x$ by $1$ .

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

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 2.6

The final solution is all the values that make $3 x e^{- 2 x} (- x + 1) = 0$ true.

$x = 0, 1$

Step 3