Question

How do you solve `x+2=e^(x)`?

Answer 1

Use Newton's Method

`x = 1.146193` and `x = -1.84141`

Explanation:

You cannot solve the equation using algebraic methods. For this type of equation, I use a numerical analysis technique called Newton's Method.

Here is a reference to Newton's method

Let `f(x) = e^x - x - 2 = 0`

`f'(x) = e^x - 1`

You start with a guess for `x_0` and then do the following computation to move closer to the solution:

`x_(n+1) = x_n - f(x_n)/(f'(x_n))`

You do computation, feeding each step back into the equation, until the number that you get doesn't change from the previous number.

Because Newton's Method is computationally intensive, I use an Excel Spreadsheet.

1. Open an Excel Spreadsheet

Into cell A1 enter your guess for `x_0`. I entered 1 into cell A1.

Into cell A2 enter the following expression:

=A1 - (EXP(A1) - A1 - 2)/(EXP(A1) - 1)

Copy the contents of cell A2 into the clipboard and then paste it into cell A3 through A10.

You will see that the number quickly converges on `x = 1.146193`

Edit: After reading a very nice comment from Shell. I decided to find the second root by changing the value of cell A1 from 1 to -1. The spreadsheet quickly converges on the value `x = -1.84141`

Answer 2

This question cannot be solved algebraically. Graphing gives `x=-1.841` and `x=1.146`.

Explanation:

The left side of the equation `x+2` is algebraic.

The right side of the equation `e^x` is transcendental (it can't be expressed as a polynomial e.g. exponentials, logs, trig functions).

This equation cannot be solved algebraically but it can be solved graphically.

To solve, plot both `color(red)(y=x+2)` and `color(blue)(y=e^x)` in a graphing utility or graphing calculator. The solutions are the `x` coordinates of the intersections.