Question 419199
Let x = length of the side of the small square.  This means that:
x + 1 = length of the side of the large square.

The area of the small square is x^2
The area of the large square is (x + 1) ^ 2

Since the combined area of the two squares is 113, we have

x^2 + ((x + 1) ^ 2) = 113
x^2 + (x^2 + 2x + 1) = 113  (expanding the expression)
2x^2 + 2x + 1 = 113   (combining like terms)
2x^2 + 2x - 112 = 0  (subtracting 113 from both sides)
x^2 + x - 56 = 0 (dividing both sides by 2)
(x - 7)(x + 8) = 0  (factoring)
x = 7  or x = -8  (setting each factor to zero and solving)

So, the side of the small square is either 7 or -8.  Since it doesn't make sense for a square to have a side -8 units long, the side of the small square is 7.

Answer: the length of the side of the small square is 7.  (And, BTW, the length of the side of the large square is x + 1 = 7 + 1 = 8).