Write a latex document with a proof of the fundamental theorem of algebra.
The Fundamental Theorem of Algebra states that if you have two algebraic equations, ax^2 + bx + c = 0 and dax^2 + ex + f = 0, where a, b, c, d, e and f are constants, then there is only one solution to this system: x = -b ± √(4aπc/d)

To prove this result, we will use mathematical induction. The basis step is to assume that there exist solutions x1 and x2 for these systems, which can be expressed as:

x1 = (-b ± √(4aπc/d)) / (2a),
x2 = (-b ± √(4aπc/d)) / (2b).

Now, we need to show that these are indeed solutions. We start by substituting them into each equation:

ax^2 + bx + c = 0 -> 2a(((-b ± √(4aπc/d)) / (2a))^2) + 4aπc/d * 1 = 0 -> 4ac - 16abπdc/(4aπcd) = 0 -> a^2 + 4adpi/(16abc)^2 - ab + 4adpi/(16abc) = 0 -> 5adpi^2/(64abc)^3 = -5abi + 4adpi^2/(64abc) = 0 -> 5api^2 - 20apib + 20apid - 15abi = 0 -> 25api^2 - 120apid + 90api = 0 -> 25a^2 - 120a + 90 = 0 -> 100 - 180 = 0 -> 50 = 0 or -20apib = 40apid.

So both of our initial assumptions are true: there do exist solutions to the system and they can be expressed as given above. Therefore, we can apply the fundamental theorem of algebra to conclude that there is only one possible solution, which must be (-b ± √(4aπc/d))/ (2a).