Precalculus Examples

$e^{x} - 6 e^{- x} - 1 = 0$

Step 1

Rewrite $e^{- x}$ as exponentiation.

$e^{x} - 6 {(e^{x})}^{- 1} - 1 = 0$

Step 2

Substitute $u$ for $e^{x}$ .

$u - 6 u^{- 1} - 1 = 0$

Step 3

Simplify each term.

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Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$u - 6 \frac{1}{u} - 1 = 0$

Step 3.2

Combine $- 6$ and $\frac{1}{u}$ .

$u + \frac{- 6}{u} - 1 = 0$

Step 3.3

Move the negative in front of the fraction.

$u - \frac{6}{u} - 1 = 0$

Step 4

Solve for $u$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, u, 1, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$u$

Step 4.2

Multiply each term in $u - \frac{6}{u} - 1 = 0$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $u - \frac{6}{u} - 1 = 0$ by $u$ .

$u \cdot u - \frac{6}{u} u - u = 0 u$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $u$ by $u$ .

$u^{2} - \frac{6}{u} u - u = 0 u$

Step 4.2.2.1.2

Cancel the common factor of $u$ .

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Step 4.2.2.1.2.1

Move the leading negative in $- \frac{6}{u}$ into the numerator.

$u^{2} + \frac{- 6}{u} u - u = 0 u$

Step 4.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 6}{u} u - u = 0 u$

Step 4.2.2.1.2.3

Rewrite the expression.

$u^{2} - 6 - u = 0 u$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $0$ by $u$ .

$u^{2} - 6 - u = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Factor $u^{2} - 6 - u$ using the AC method.

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Step 4.3.1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 4.3.1.2

Write the factored form using these integers.

$(u - 3) (u + 2) = 0$

Step 4.3.2

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

$u - 3 = 0$

$u + 2 = 0$

Step 4.3.3

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 4.3.3.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 4.3.3.2

Add $3$ to both sides of the equation.

$u = 3$

Step 4.3.4

Set $u + 2$ equal to $0$ and solve for $u$ .

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Step 4.3.4.1

Set $u + 2$ equal to $0$ .

$u + 2 = 0$

Step 4.3.4.2

Subtract $2$ from both sides of the equation.

$u = - 2$

Step 4.3.5

The final solution is all the values that make $(u - 3) (u + 2) = 0$ true.

$u = 3, - 2$

Step 5

Substitute $3$ for $u$ in $u = e^{x}$ .

$3 = e^{x}$

Step 6

Solve $3 = e^{x}$ .

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Step 6.1

Rewrite the equation as $e^{x} = 3$ .

$e^{x} = 3$

Step 6.2

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

$\ln (e^{x}) = \ln (3)$

Step 6.3

Expand the left side.

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Step 6.3.1

Expand $\ln (e^{x})$ by moving $x$ outside the logarithm.

$x \ln (e) = \ln (3)$

Step 6.3.2

The natural logarithm of $e$ is $1$ .

$x \cdot 1 = \ln (3)$

Step 6.3.3

Multiply $x$ by $1$ .

$x = \ln (3)$

Step 7

Substitute $- 2$ for $u$ in $u = e^{x}$ .

$- 2 = e^{x}$

Step 8

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

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Step 8.1

Rewrite the equation as $e^{x} = - 2$ .

$e^{x} = - 2$

Step 8.2

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

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

Step 8.3

The equation cannot be solved because $\ln (- 2)$ is undefined.

Undefined

Step 8.4

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

No solution

Step 9

List the solutions that makes the equation true.

$x = \ln (3)$

Step 10