Calculus Examples

$x^{2} e^{2 x} + 2 x e^{2 x} = 8 e^{2 x}$

Step 1

Subtract $8 e^{2 x}$ from both sides of the equation.

$x^{2} e^{2 x} + 2 x e^{2 x} - 8 e^{2 x} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $e^{2 x}$ out of $x^{2} e^{2 x} + 2 x e^{2 x} - 8 e^{2 x}$ .

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Step 2.1.1

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

$e^{2 x} x^{2} + 2 x e^{2 x} - 8 e^{2 x} = 0$

Step 2.1.2

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

$e^{2 x} x^{2} + e^{2 x} (2 x) - 8 e^{2 x} = 0$

Step 2.1.3

Factor $e^{2 x}$ out of $- 8 e^{2 x}$ .

$e^{2 x} x^{2} + e^{2 x} (2 x) + e^{2 x} \cdot - 8 = 0$

Step 2.1.4

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

$e^{2 x} (x^{2} + 2 x) + e^{2 x} \cdot - 8 = 0$

Step 2.1.5

Factor $e^{2 x}$ out of $e^{2 x} (x^{2} + 2 x) + e^{2 x} \cdot - 8$ .

$e^{2 x} (x^{2} + 2 x - 8) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 2.2.1.2

Write the factored form using these integers.

$e^{2 x} ((x - 2) (x + 4)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$e^{2 x} (x - 2) (x + 4) = 0$

Step 3

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

$e^{2 x} = 0$

$x - 2 = 0$

$x + 4 = 0$

Step 4

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

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Step 4.1

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

$e^{2 x} = 0$

Step 4.2

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

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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.

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

Step 4.2.2

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

Undefined

Step 4.2.3

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

No solution

Step 5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

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Step 6.1

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

$x + 4 = 0$

Step 6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7

The final solution is all the values that make $e^{2 x} (x - 2) (x + 4) = 0$ true.

$x = 2, - 4$