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

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:

  • a - b = 5
  • (a + b)^2 = 169
  • 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)

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