Precalculus Examples

$10 e^{x} - 15 - 45 e^{- x} = 0$

Step 1

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

$10 e^{x} - 15 - 45 {(e^{x})}^{- 1} = 0$

Step 2

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

$10 u - 15 - 45 u^{- 1} = 0$

Step 3

Simplify each term.

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Step 3.1

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

$10 u - 15 - 45 \frac{1}{u} = 0$

Step 3.2

Combine $- 45$ and $\frac{1}{u}$ .

$10 u - 15 + \frac{- 45}{u} = 0$

Step 3.3

Move the negative in front of the fraction.

$10 u - 15 - \frac{45}{u} = 0$

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.

$1, 1, u, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$u$

Step 4.2

Multiply each term in $10 u - 15 - \frac{45}{u} = 0$ by $u$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $10 u - 15 - \frac{45}{u} = 0$ by $u$ .

$10 u \cdot u - 15 u - \frac{45}{u} u = 0 u$

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 $u$ by adding the exponents.

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Step 4.2.2.1.1.1

Move $u$ .

$10 (u \cdot u) - 15 u - \frac{45}{u} u = 0 u$

Step 4.2.2.1.1.2

Multiply $u$ by $u$ .

$10 u^{2} - 15 u - \frac{45}{u} u = 0 u$

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 $- \frac{45}{u}$ into the numerator.

$10 u^{2} - 15 u + \frac{- 45}{u} u = 0 u$

Step 4.2.2.1.2.2

Cancel the common factor.

$10 u^{2} - 15 u + \frac{- 45}{u} u = 0 u$

Step 4.2.2.1.2.3

Rewrite the expression.

$10 u^{2} - 15 u - 45 = 0 u$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $0$ by $u$ .

$10 u^{2} - 15 u - 45 = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Factor the left side of the equation.

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Step 4.3.1.1

Factor $5$ out of $10 u^{2} - 15 u - 45$ .

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Step 4.3.1.1.1

Factor $5$ out of $10 u^{2}$ .

$5 (2 u^{2}) - 15 u - 45 = 0$

Step 4.3.1.1.2

Factor $5$ out of $- 15 u$ .

$5 (2 u^{2}) + 5 (- 3 u) - 45 = 0$

Step 4.3.1.1.3

Factor $5$ out of $- 45$ .

$5 (2 u^{2}) + 5 (- 3 u) + 5 (- 9) = 0$

Step 4.3.1.1.4

Factor $5$ out of $5 (2 u^{2}) + 5 (- 3 u)$ .

$5 (2 u^{2} - 3 u) + 5 (- 9) = 0$

Step 4.3.1.1.5

Factor $5$ out of $5 (2 u^{2} - 3 u) + 5 (- 9)$ .

$5 (2 u^{2} - 3 u - 9) = 0$

Step 4.3.1.2

Factor.

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Step 4.3.1.2.1

Factor by grouping.

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Step 4.3.1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 9 = - 18$ and whose sum is $b = - 3$ .

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Step 4.3.1.2.1.1.1

Factor $- 3$ out of $- 3 u$ .

$5 (2 u^{2} - 3 u - 9) = 0$

Step 4.3.1.2.1.1.2

Rewrite $- 3$ as $3$ plus $- 6$

$5 (2 u^{2} + (3 - 6) u - 9) = 0$

Step 4.3.1.2.1.1.3

Apply the distributive property.

$5 (2 u^{2} + 3 u - 6 u - 9) = 0$

Step 4.3.1.2.1.2

Factor out the greatest common factor from each group.

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Step 4.3.1.2.1.2.1

Group the first two terms and the last two terms.

$5 ((2 u^{2} + 3 u) - 6 u - 9) = 0$

Step 4.3.1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$5 (u (2 u + 3) - 3 (2 u + 3)) = 0$

Step 4.3.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 3$ .

$5 ((2 u + 3) (u - 3)) = 0$

Step 4.3.1.2.2

Remove unnecessary parentheses.

$5 (2 u + 3) (u - 3) = 0$

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 $0$ .

$2 u + 3 = 0$

$u - 3 = 0$

Step 4.3.3

Set $2 u + 3$ equal to $0$ and solve for $u$ .

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Step 4.3.3.1

Set $2 u + 3$ equal to $0$ .

$2 u + 3 = 0$

Step 4.3.3.2

Solve $2 u + 3 = 0$ for $u$ .

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Step 4.3.3.2.1

Subtract $3$ from both sides of the equation.

$2 u = - 3$

Step 4.3.3.2.2

Divide each term in $2 u = - 3$ by $2$ and simplify.

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Step 4.3.3.2.2.1

Divide each term in $2 u = - 3$ by $2$ .

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 4.3.3.2.2.2

Simplify the left side.

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Step 4.3.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 4.3.3.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 3}{2}$

Step 4.3.3.2.2.3

Simplify the right side.

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Step 4.3.3.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{3}{2}$

Step 4.3.4

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

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Step 4.3.4.1

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

$u - 3 = 0$

Step 4.3.4.2

Add $3$ to both sides of the equation.

$u = 3$

Step 4.3.5

The final solution is all the values that make $5 (2 u + 3) (u - 3) = 0$ true.

$u = - \frac{3}{2}, 3$

Step 5

Substitute $- \frac{3}{2}$ for $u$ in $u = e^{x}$ .

$- \frac{3}{2} = e^{x}$

Step 6

Solve $- \frac{3}{2} = e^{x}$ .

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Step 6.1

Rewrite the equation as $e^{x} = - \frac{3}{2}$ .

$e^{x} = - \frac{3}{2}$

Step 6.2

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

$\ln (e^{x}) = \ln (- \frac{3}{2})$

Step 6.3

The equation cannot be solved because $\ln (- \frac{3}{2})$ is undefined.

Undefined

Step 6.4

There is no solution for $e^{x} = - \frac{3}{2}$

No solution

Step 7

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

$3 = e^{x}$

Step 8

Solve $3 = e^{x}$ .

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Step 8.1

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

$e^{x} = 3$

Step 8.2

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

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

Step 8.3

Expand the left side.

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Multiply $x$ by $1$ .

$x = \ln (3)$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \ln (3)$

Decimal Form:

$x = 1.09861228 \dots$