1) To solve the equation x^2 + 7x = 0, we first factor out x:

x(x + 7) = 0

Now we have two possible solutions:

x = 0 or x + 7 = 0
x = 0 or x = -7

Therefore, the solutions to the equation x^2 + 7x = 0 are x = 0 and x = -7.

2) To solve the equation x^2 - 25 = 0, we can factor the equation as the difference of squares:

(x + 5)(x - 5) = 0

Now we have two possible solutions:

x + 5 = 0 or x - 5 = 0
x = -5 or x = 5

Therefore, the solutions to the equation x^2 - 25 = 0 are x = -5 and x = 5.