Solve for x 6e^x-4e^(-x)=5

$6 e^{x} - 4 e^{- x} = 5$

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

$6 e^{x} - 4 {(e^{x})}^{- 1} = 5$

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

$6 u - 4 u^{- 1} = 5$

Simplify each term.

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

$6 u - 4 \frac{1}{u} = 5$

Combine $- 4$ and $\frac{1}{u}$ .

$6 u + \frac{- 4}{u} = 5$

Move the negative in front of the fraction.

$6 u - \frac{4}{u} = 5$

Solve for $u$ .

Find the LCD of the terms in the equation.

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

$1, u, 1$

The LCM of one and any expression is the expression.

$u$

Multiply each term in $6 u - \frac{4}{u} = 5$ by $u$ to eliminate the fractions.

Multiply each term in $6 u - \frac{4}{u} = 5$ by $u$ .

$6 u \cdot u - \frac{4}{u} u = 5 u$

Simplify the left side.

Simplify each term.

Multiply $u$ by $u$ by adding the exponents.

Move $u$ .

$6 (u \cdot u) - \frac{4}{u} u = 5 u$

Multiply $u$ by $u$ .

$6 u^{2} - \frac{4}{u} u = 5 u$

Cancel the common factor of $u$ .

Move the leading negative in $- \frac{4}{u}$ into the numerator.

$6 u^{2} + \frac{- 4}{u} u = 5 u$

Cancel the common factor.

$6 u^{2} + \frac{- 4}{u} u = 5 u$

Rewrite the expression.

$6 u^{2} - 4 = 5 u$

Solve the equation.

Subtract $5 u$ from both sides of the equation.

$6 u^{2} - 4 - 5 u = 0$

Factor by grouping.

Reorder terms.

$6 u^{2} - 5 u - 4 = 0$

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 $a \cdot c = 6 \cdot - 4 = - 24$ and whose sum is $b = - 5$ .

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

$6 u^{2} - 5 u - 4 = 0$

Rewrite $- 5$ as $3$ plus $- 8$

$6 u^{2} + (3 - 8) u - 4 = 0$

Apply the distributive property.

$6 u^{2} + 3 u - 8 u - 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 u^{2} + 3 u) - 8 u - 4 = 0$

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

$3 u (2 u + 1) - 4 (2 u + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$(2 u + 1) (3 u - 4) = 0$

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

$2 u + 1 = 0$

$3 u - 4 = 0$

Set $2 u + 1$ equal to $0$ and solve for $u$ .

Set $2 u + 1$ equal to $0$ .

$2 u + 1 = 0$

Solve $2 u + 1 = 0$ for $u$ .

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Divide each term in $2 u = - 1$ by $2$ and simplify.

Divide each term in $2 u = - 1$ by $2$ .

$\frac{2 u}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 1}{2}$

Divide $u$ by $1$ .

$u = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$u = - \frac{1}{2}$

Set $3 u - 4$ equal to $0$ and solve for $u$ .

Set $3 u - 4$ equal to $0$ .

$3 u - 4 = 0$

Solve $3 u - 4 = 0$ for $u$ .

Add $4$ to both sides of the equation.

$3 u = 4$

Divide each term in $3 u = 4$ by $3$ and simplify.

Divide each term in $3 u = 4$ by $3$ .

$\frac{3 u}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 u}{3} = \frac{4}{3}$

Divide $u$ by $1$ .

$u = \frac{4}{3}$

The final solution is all the values that make $(2 u + 1) (3 u - 4) = 0$ true.

$u = - \frac{1}{2}, \frac{4}{3}$

Substitute $- \frac{1}{2}$ for $u$ in $u = e^{x}$ .

$- \frac{1}{2} = e^{x}$

Solve $- \frac{1}{2} = e^{x}$ .

Rewrite the equation as $e^{x} = - \frac{1}{2}$ .

$e^{x} = - \frac{1}{2}$

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

$\ln (e^{x}) = \ln (- \frac{1}{2})$

The equation cannot be solved because $\ln (- \frac{1}{2})$ is undefined.

Undefined

There is no solution for $e^{x} = - \frac{1}{2}$

No solution

Substitute $\frac{4}{3}$ for $u$ in $u = e^{x}$ .

$\frac{4}{3} = e^{x}$

Solve $\frac{4}{3} = e^{x}$ .

Rewrite the equation as $e^{x} = \frac{4}{3}$ .

$e^{x} = \frac{4}{3}$

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

$\ln (e^{x}) = \ln (\frac{4}{3})$

Expand the left side.

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

$x \ln (e) = \ln (\frac{4}{3})$

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

$x \cdot 1 = \ln (\frac{4}{3})$

Multiply $x$ by $1$ .

$x = \ln (\frac{4}{3})$

The result can be shown in multiple forms.

Exact Form:

$x = \ln (\frac{4}{3})$

Decimal Form:

$x = 0.28768207 \dots$

Solve for x 6e^x-4e^(-x)=5

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