The difference between two positive numbers is 5 and the square of their sum is 169

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The difference between two positive numbers is 5 and the square of their sum is 169.

Let the two positive numbers be a and b. We have the following equations:

1. a - b = 5
2. (a + b)^2 = 169
3. Rearrange (1) by adding b to each side. We have a = b + 5

Now substitute (3) into (2):
(b + 5 + b)^2 = 169
(2b + 5)^2 = 169

Run (2b + 5)^2 through our search engine, and you get:
4b^2 + 20b + 25

Set this equal to 169 above:
4b^2 + 20b + 25 = 169

Run that quadratic equation in our search engine, and you get:
b = (-9, 4)

But the problem asks for positive numbers. So b = 4 is one of our solutions.

Substitute b = 4 into equation (1) above, and we get:
a - b = 5
a - 4 = 5
a = 9

Therefore, we have (a, b) = (9, 4)