1. Indeed, 5^k - 5^(k-1) = 5^k - (1/5)*(5^k). What you're actually doing here is spotting a common factor, which is 5^k. You can easily factorize it now, since it will be obvious that:

5^k - (1/5)*(5^k) = (5^k)*(1 - 1/5) = (5^k)*4/5

BUT

you made a mistake here:

5^(k+1) = 25^k

5^(k+1) = 5*(5^k) - as you can see, this is very different from 25^k. Just consider k = 3: 5^(k+1) = 5^4 = 625 = 5*125 = 5*5^3. HOWEVER, 25^4 is actually 625*625, since 25^2 = 625.


2. It's actually the easiest thing to know that the equation has only one root. This only happens when you can make a perfect square out of it, in other words when it follows the pattern on the formula:

(x + n)^2 = x^2 + 2xn + n^2.

Let me give you a few examples:
x^2 + 4x + 4 = x^2 + 2*2*x + 2^2 = (x + 2)^2 - the only root will be -2

x^2 + 10x + 25 = (x + 5)^2 - the only root will be -5

x^2 - 6x + 9 = (x - 3)^2 - the only root will be 3

x^2 - 4x + 3 - we can't find a square here, because x^2 - 4x + 3 = (x - 1)(x - 3) - as you can see, the equation has two roots, 1 and 3.