Solve for x e^x-20e^(-x)-1=0

Problem

ex−20*e(−x)−1=0

Solution

1. Multiply the entire equation by ex to eliminate the negative exponent.

ex*(ex−20*e(−x)−1)=0⋅ex

(ex)2−20−ex=0

2. Rearrange the terms into a standard quadratic form.

(ex)2−ex−20=0

3. Substitute a new variable u=ex to make the equation easier to solve.

u2−u−20=0

4. Factor the quadratic equation by finding two numbers that multiply to −20 and add to −1

(u−5)*(u+4)=0

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

u=5

u=−4

6. Back-substitute ex for u and solve for x

ex=5⇒x=ln(5)

ex=−4

7. Discard the extraneous solution because ex must always be greater than zero for real values of x

ex=−4* has no real solution

Final Answer

x=ln(5)

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