Let's start how to find the value of the infinite repeating fraction $$1 + \cfrac{2}{1 + \cfrac{2}{1 + \cdots}}$$
We set the entire expression to $x$, and we notice that the entire expression under the fraction is equal to $x$. This gives the equation $x = 1 + \frac{2}{x}$ We multiply both sides by $x$ and move everything to the left to get $x^2 - x - 2 = 0$. We factor the quadratic as $(x - 2)(x + 1)$, which gives $x = 2$ and $x = -1$. Obviously, the fraction cannot be equal to -1. So, the fraction is equal to 2.
When we try to do the same on $$7 - \cfrac{12}{7 - \cfrac{12}{\cdots}}$$ we get the quadratic $x^2 - 7x + 12 = 0$ which leads to the solutions $x = 3$ and $x = 4$. But this is an expression, and it clearly cannot be equal to 2 different values!
Which solution is incorrect, and why?