Calculus Examples

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

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

Step 1

Reorder factors in

2 e^{x} x + e^{x} x^{2}

$2 e^{x} x + e^{x} x^{2}$ .

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

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

Step 2

Factor

x e^{x}

$x e^{x}$ out of

2 x e^{x} + x^{2} e^{x}

$2 x e^{x} + x^{2} e^{x}$ .

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Step 2.1

Factor

x e^{x}

$x e^{x}$ out of

2 x e^{x}

$2 x e^{x}$ .

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

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

Step 2.2

Factor

x e^{x}

$x e^{x}$ out of

x^{2} e^{x}

$x^{2} e^{x}$ .

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

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

Step 2.3

Factor

x e^{x}

$x e^{x}$ out of

x e^{x} (2) + x e^{x} (x)

$x e^{x} (2) + x e^{x} (x)$ .

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

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

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

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

Step 3

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

x = 0

$x = 0$

e^{x} = 0

$e^{x} = 0$

2 + x = 0

$2 + x = 0$

Step 4

Set

x

$x$ equal to

0

$0$ .

x = 0

$x = 0$

Step 5

Set

e^{x}

$e^{x}$ equal to

0

$0$ and solve for

x

$x$ .

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Step 5.1

Set

e^{x}

$e^{x}$ equal to

0

$0$ .

e^{x} = 0

$e^{x} = 0$

Step 5.2

Solve

e^{x} = 0

$e^{x} = 0$ for

x

$x$ .

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Step 5.2.1

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

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

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

Step 5.2.2

The equation cannot be solved because

\ln (0)

$\ln (0)$ is undefined.

Undefined

Step 5.2.3

There is no solution for

e^{x} = 0

$e^{x} = 0$

No solution

Step 6

Set

2 + x

$2 + x$ equal to

0

$0$ and solve for

x

$x$ .

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Step 6.1

Set

2 + x

$2 + x$ equal to

0

$0$ .

2 + x = 0

$2 + x = 0$

Step 6.2

Subtract

2

$2$ from both sides of the equation.

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 7

The final solution is all the values that make

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

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

x = 0, - 2

$x = 0, - 2$