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14 and 10
Algebraically:
xy = 140
x + y = 24 yields y = 24 - x
xy = 140 replace y with 24 - x
x(24 - x) = 140 distribute x on the left side
-x2 + 24x = 140 multiply by -1 both sides
x2 - 24x = -140 complete the square by adding 122 to both sides
x2 - 24x + 122 = -140 + 144
(x - 12)2 = 4 take the square root of both sides
x - 12 = ± 2 add 12 to both sides
x = 12 ± 2
x = 12 + 2 or x = 12 - 2
x = 14 or x = 10
Thus, the numbers are 14 and 10.
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