### Instruction: Find the sum of all real values of $x$ that satisfy
\[x = 1 - x + x^2 - x^3 + x^4 - x^5 + \dotsb.\].### Response: This is a very interesting problem.
It looks like an infinite series, but it is also an equation with $x$.
I wonder if there is a way to simplify this series.
I notice that it has alternating signs and powers of $x$.
Maybe I can factor out a $x$ from every term except the first one.
That would give me
\[x = 1 - x(1 - x + x^2 - x^3 + x^4 - \dotsb).\]
Hey, this looks familiar.
The expression inside the parentheses is the same as the original series.
So I can substitute $x$ for it.
That gives me
\[x = 1 - x^2.\]
Now this is a simple quadratic equation.
I can rearrange it and factor it as
\[x^2 + x - 1 = 0.\]
To find the solutions, I can use the quadratic formula.
That gives me
\[x = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2}.\]
These are the two possible values of $x$ that satisfy the original equation.
To find the sum of them, I just need to add them together.