6^(1+x)+6^(1-x)=37

6*6^(x) + 6*6^(-x) = 37

6^(x) + 6^(-x) = 37/6

6^x + 1/6^(x) = 37/6

6^x*6^x + 1 = (37/6)*6^x

6^(2x) - (37/6)*6^(x) + 1 = 0

[6^(x)]^2 - (37/6)*[6^(x)] + 1 = 0

Let y = 6^x

y^2 - (37/6)y + 1 = 0

Factor, complete the square, use the quadratic formula, whatever. You get:

y = 1/6 or y = 6.

Thus

6^x = 1/6 ==> x = -1

or

6^x = 6 ==> x = 1

Thus x = -1 or x = 1.

-Dan