solving for x--------- x +2 = cosh x?

This cannot be solved exactly by algebra. You need to use a numerical method.

Try an iterative formula.

x + 2 = cosh x = (e^x + e^-x)/2

x(n+1) = (e^xn + e^-xn)/2 - 2

Make a first guess (x1 = 1) and use this to repeatedly improve it, however, this converges slowly.

Alternatively, x + 2 = (e^x + e^-x)/2

2x + 4 = e^x + e^-x

2x + 4 - e^-x = e^x

x(n+1) = ln(2xn + 4 - e^-xn)

I think that this is faster.

Sometimes a different first guess will give an another solution but sometimes a different iterative formula is needed.

x= cosh(x)-2

x = (e^x + e^-x)/2 -2

2x = e^x + e^-x -4

2x-e^x-e^-x+4=0

Graph y= 2x-e^x-e^-x+4=0 and see where the graph crosses the x-axis.

You may use a graphic calculator or use numerical methods.

x= 2.08518601 (Newton's method)

x= -0.72526372 (Newton's method)

There are two solutions.

x^2-25=x^2-5x 5x=25 x=5

coshx=1/2(e^x+e^-x)

2x+4=e^x+e^-x

the sol num is

x ~~ -0.7252637249181236

x ~~ 2.085186014173246