# Solve for x e^x-6e^(-x)-1=0

Rewrite as .

Rewrite as .

Substitute for .

Simplify each term.

Rewrite the expression using the negative exponent rule .

Combine and .

Move the negative in front of the fraction.

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.

Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for is itself.

occurs time.

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply each term by and simplify.

Multiply each term in by in order to remove all the denominators from the equation.

Simplify each term.

Multiply by .

Cancel the common factor of .

Move the leading negative in into the numerator.

Cancel the common factor.

Rewrite the expression.

Rewrite as .

Multiply by .

Solve the equation.

Factor using the AC method.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

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

Set the first factor equal to and solve.

Set the first factor equal to .

Add to both sides of the equation.

Set the next factor equal to and solve.

Set the next factor equal to .

Subtract from both sides of the equation.

The final solution is all the values that make true.

Substitute for in .

Rewrite the equation as .

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

Expand the left side.

Expand by moving outside the logarithm.

The natural logarithm of is .

Multiply by .

Substitute for in .

Rewrite the equation as .

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

The equation cannot be solved because is undefined.

Undefined

There is no solution for

No solution

No solution

List the solutions that makes the equation true.

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

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