OK. We start with ex+e-x=6.
I multiplied through by ex. This led to
e2x - 6ex + 1 = 0
Then I decided to set y = ex. So now it's
y2 - 6y + 1 = 0
This you solve with the Quadratic Formula and get
y = (6 ± √(36 - 4))/2 or y = 5.8 or 0.2 approx.
So ex = 5.8 or 0.2. This means x = 1.76 or -1.61.
Sanity check: I ran this problem through Wolfram Alpha and got ±1.76. This is why you run sanity checks. No doubt I made a rounding error on the negative value. I apologize for that.