Solve the equation algebraically?

Question

e^x+e^-x=3

Answer 1

Let y = e^x. Then e^-x = 1/y, and the equation becomes

y + (1/y) = 3

y² + 1 = 3y

y² - 3y + 1 = 0

y = [-(-3) ± √(9 - 4)]/2 = (3 ± √5)/2

y = e^x ⇒ x = ln(y)

Therefore, the solutions are

x = ln[(3 + √5)/2] and x = ln[(3 - √5)/2]