Finite Math Examples

$y = 4 x^{3} e^{x} + x^{4} e^{x}$

Step 1

Set $4 x^{3} e^{x} + x^{4} e^{x}$ equal to $0$ .

$4 x^{3} e^{x} + x^{4} e^{x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{3} e^{x}$ out of $4 x^{3} e^{x} + x^{4} e^{x}$ .

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Step 2.1.1

Factor $x^{3} e^{x}$ out of $4 x^{3} e^{x}$ .

$x^{3} e^{x} (4) + x^{4} e^{x} = 0$

Step 2.1.2

Factor $x^{3} e^{x}$ out of $x^{4} e^{x}$ .

$x^{3} e^{x} (4) + x^{3} e^{x} (x) = 0$

Step 2.1.3

Factor $x^{3} e^{x}$ out of $x^{3} e^{x} (4) + x^{3} e^{x} (x)$ .

$x^{3} e^{x} (4 + x) = 0$

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 $0$ .

$x^{3} = 0$

$e^{x} = 0$

$4 + x = 0$

Step 2.3

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.3.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 2.3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 2.4

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

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Step 2.4.1

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

$e^{x} = 0$

Step 2.4.2

Solve $e^{x} = 0$ 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.

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

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 $e^{x} = 0$

No solution

Step 2.5

Set $4 + x$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 + x$ equal to $0$ .

$4 + x = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

The final solution is all the values that make $x^{3} e^{x} (4 + x) = 0$ true.

$x = 0, - 4$

Step 3