Precalculus Examples

e^{x} - 30 e^{- x} - 1 = 0

$e^{x} - 30 e^{- x} - 1 = 0$

Step 1

Rewrite

e^{- x}

$e^{- x}$ as exponentiation.

e^{x} - 30 {(e^{x})}^{- 1} - 1 = 0

$e^{x} - 30 {(e^{x})}^{- 1} - 1 = 0$

Step 2

Substitute

u

$u$ for

e^{x}

$e^{x}$ .

u - 30 u^{- 1} - 1 = 0

$u - 30 u^{- 1} - 1 = 0$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

u - 30 \frac{1}{u} - 1 = 0

$u - 30 \frac{1}{u} - 1 = 0$

Step 3.2

Combine

- 30

$- 30$ and

\frac{1}{u}

$\frac{1}{u}$ .

u + \frac{- 30}{u} - 1 = 0

$u + \frac{- 30}{u} - 1 = 0$

Step 3.3

Move the negative in front of the fraction.

u - \frac{30}{u} - 1 = 0

$u - \frac{30}{u} - 1 = 0$

u - \frac{30}{u} - 1 = 0

$u - \frac{30}{u} - 1 = 0$

Step 4

Solve for

u

$u$ .

Tap for more steps...

Step 4.1

Find the LCD of the terms in the equation.

Tap for more steps...

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

$1, u, 1, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

u

$u$

u

$u$

Step 4.2

Multiply each term in

u - \frac{30}{u} - 1 = 0

$u - \frac{30}{u} - 1 = 0$ by

u

$u$ to eliminate the fractions.

Tap for more steps...

Step 4.2.1

Multiply each term in

u - \frac{30}{u} - 1 = 0

$u - \frac{30}{u} - 1 = 0$ by

u

$u$ .

u \cdot u - \frac{30}{u} u - u = 0 u

$u \cdot u - \frac{30}{u} u - u = 0 u$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Multiply

u

$u$ by

u

$u$ .

u^{2} - \frac{30}{u} u - u = 0 u

$u^{2} - \frac{30}{u} u - u = 0 u$

Step 4.2.2.1.2

Cancel the common factor of

u

$u$ .

Tap for more steps...

Step 4.2.2.1.2.1

Move the leading negative in

- \frac{30}{u}

$- \frac{30}{u}$ into the numerator.

u^{2} + \frac{- 30}{u} u - u = 0 u

$u^{2} + \frac{- 30}{u} u - u = 0 u$

Step 4.2.2.1.2.2

Cancel the common factor.

$u^{2} + \frac{- 30}{u} u - u = 0 u$

Step 4.2.2.1.2.3

Rewrite the expression.

$u^{2} - 30 - u = 0 u$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Multiply

$0$ by

$u$ .

$u^{2} - 30 - u = 0$

Step 4.3

Solve the equation.

Tap for more steps...

Step 4.3.1

Factor

$u^{2} - 30 - u$ using the AC method.

Tap for more steps...

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

$- 30$ and whose sum is

$- 1$ .

$- 6, 5$

Step 4.3.1.2

Write the factored form using these integers.

$(u - 6) (u + 5) = 0$

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

$0$ .

$u - 6 = 0$

$u + 5 = 0$

Step 4.3.3

Set

$u - 6$ equal to

$0$ and solve for

$u$ .

Tap for more steps...

Step 4.3.3.1

Set

$u - 6$ equal to

$0$ .

$u - 6 = 0$

Step 4.3.3.2

Add

$6$ to both sides of the equation.

$u = 6$

Step 4.3.4

Set

$u + 5$ equal to

$0$ and solve for

$u$ .

Tap for more steps...

Step 4.3.4.1

Set

$u + 5$ equal to

$0$ .

$u + 5 = 0$

Step 4.3.4.2

Subtract

$5$ from both sides of the equation.

$u = - 5$

Step 4.3.5

The final solution is all the values that make

$(u - 6) (u + 5) = 0$ true.

$u = 6, - 5$

Step 5

Substitute

$6$ for

$u$ in

$u = e^{x}$ .

$6 = e^{x}$

Step 6

Solve

$6 = e^{x}$ .

Tap for more steps...

Step 6.1

Rewrite the equation as

$e^{x} = 6$ .

$e^{x} = 6$

Step 6.2

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

$\ln (e^{x}) = \ln (6)$

Step 6.3

Expand the left side.

Tap for more steps...

Step 6.3.1

Expand

$\ln (e^{x})$ by moving

$x$ outside the logarithm.

$x \ln (e) = \ln (6)$

Step 6.3.2

The natural logarithm of

$e$ is

$1$ .

$x \cdot 1 = \ln (6)$

Step 6.3.3

Multiply

$x$ by

$1$ .

$x = \ln (6)$

Step 7

Substitute

$- 5$ for

$u$ in

$u = e^{x}$ .

$- 5 = e^{x}$

Step 8

Solve

$- 5 = e^{x}$ .

Tap for more steps...

Step 8.1

Rewrite the equation as

$e^{x} = - 5$ .

$e^{x} = - 5$

Step 8.2

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

$\ln (e^{x}) = \ln (- 5)$

Step 8.3

The equation cannot be solved because

$\ln (- 5)$ is undefined.

Undefined

Step 8.4

There is no solution for

$e^{x} = - 5$

No solution

Step 9

List the solutions that makes the equation true.

$x = \ln (6)$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \ln (6)$

Decimal Form:

$x = 1.79175946 \dots$