So, Im trying to make up this theory of trying to solve any equation ever using recurrences... Ill show you what I mean Consider the quadratic function - f(x) = x^2 - 3x + 2 Well, obviously you could do this the easy way and do factoring and figure out x = 2 or 1... no big deal right? But what if we try a different way... albeit harder and more complicated... but Itll make a bit more sense later in this post So what if we do this x^2 - 3x + 2 = 0 x(x) - 3x + 2 = 0 x(x) = 3x-2 x = (3x-2)/x So what does this mean? Well, plug in a random number into your calculator... say 10... Now do (3x-2)/x, and youll get 2.8... now (if your using a TI-84 like me), plug in ((3*Ans)-2)/Ans So Ans = 2.8, this should yield 2.28 Now Ans = 2.28, this should yield 2.125 From 2.125, you get 2.0588.... Basically, If you do this an infinite amount of times, it will converge to 2... which is one of the solutions of the equation... Now say you rearrange the equation differently... x^2 -3x+2 = 0 x(x) - 3x + 2 = 0 (x-3)x = -2 x = (-2)/(x-3) If you do the same method as above, it will converge to 1 for any initial value you put in for x, which is the other zero for the quadratic equation... So... why am I wasting my time with this? Well, consider this equation 4x = e^(.5x) Well... idk how to solve this equation.... but what if we do the thing I just did, where we set one side of the equation to x then infinitely compute the other side... x = e^(.5x) ln(4x) = .5x 2ln(4x) = x If you pick 20 for the first x, 2ln(4x) will yield 8.76, then 7.11, then 6.69, and so on, until it converges to a number 6.523371369... Is this right? well 4(x) = e^(.5x) 4(6.523371369) = e^(.5(6.523371369)) 26.09 = e^(3.26) 26.09 = 26.09 So... thats the solution.... by using an infinite recursion of setting one side of the equation to x, then we will get some sort of solution.... Now... this doesnt always work... for example, say we did (from the previous equation) 4x = e^(.5x) x = (e^(.5x))/4 If you do this, then x will converge to infinity. Now you may think that this is wrong... but I feel this is right, since they both technically (intersect) at infinity. In theory, 4(infinity) = e^(infinity).... so is it really wrong? I dont believe so.... Anyway... my theory is that you can solve a solution for an equation when you set the equation to the form of x = f(x) Using an infinite recursive calculation. Now, I really don't know how to dive more into this theory, since I think this involves math thats super crazy (more than I know). Im a computer engineer, so I dont have the biggest arsenal of theoretical math at my disposal. I do know that this might have to do something with damping, since some of the recurrences will have an overdamped, underdamped, and critically damped response to converging to the solution... but... i dont know... So... any thoughts on this? Any suggestions on how to move forward with this? Or has someone done this before? And no, this does not have anything to do with homework, this is just fun theoretical stuff Im doing on my own and I would like input/other ideas to help me move forward with this. If this post is deemed by the admins to not belong here, then Ill delete this post (no questions asked) and move it to the Homework Section (or wherever they tell me to put it). All I want is some outside input for this.