Solve for x (e^x+e^(-x))/3=10

Problem

(ex+e(−x))/3=10

Solution

1. Multiply both sides by 3 to isolate the sum of the exponential terms.

(ex+e(−x))/3=10

ex+e(−x)=30

2. Eliminate the negative exponent by multiplying every term in the equation by ex

ex*(ex+e(−x))=30*(ex)

(ex)2+1=30*ex

3. Rearrange the equation into a standard quadratic form by subtracting 30*ex from both sides.

(ex)2−30*ex+1=0

4. Substitute a temporary variable u=ex to treat the equation as a quadratic in terms of u

u2−30*u+1=0

5. Apply the quadratic formula u=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−30 and c=1

u=(30±√(,(−30)2−4*(1)*(1)))/(2*(1))

u=(30±√(,900−4))/2

u=(30±√(,896))/2

6. Simplify the square root by factoring out the largest perfect square, which is 64

√(,896)=√(,64⋅14)=8√(,14)

u=(30±8√(,14))/2

u=15±4√(,14)

7. Solve for x by substituting ex back in for u and taking the natural logarithm of both sides.

ex=15±4√(,14)

x=ln(15±4√(,14))

Final Answer

(ex+e(−x))/3=10⇒x=ln(15±4√(,14))

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