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

Problem

e^x−6*e^(−x)−1=0

Solution

1. Multiply the entire equation by e^x to eliminate the negative exponent, noting that e^x⋅e^(−x)=1

e^x*(e^x−6*e^(−x)−1)=0

(e^x)^2−e^x−6=0

2. Substitute a new variable u=e^x to transform the equation into a quadratic form.

u^2−u−6=0

3. Factor the quadratic equation by finding two numbers that multiply to −6 and add to −1

(u−3)*(u+2)=0

4. Solve for u by setting each factor to zero.

u=3

u=−2

5. Back-substitute e^x for u and solve for x using natural logarithms.

e^x=3⇒x=ln(3)

e^x=−2

6. Discard the extraneous solution because e^x must always be greater than zero for real values of x

e^x=−2⇒No real solution

Final Answer

x=ln(3)

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