Find two numbers

Let the first number be x and the second number is 27 − x.
[As the sum of both the numbers is 27]

Therefore, their product = x (27 − x)

It is given that the product of these numbers is 182.

 x (27 – x) = 182

- x² + 27x - 182 = 0
Changing the signs on both sides we get,
x² - 27x + 182 = 0
Factorizing we get , 13 and 14 are the numbers whose sum is 27 and product is 182

 x² – 13x – 14x + 182 = 0

= x(x – 13) – 14 (x – 13)= 0

= (x – 13) (x – 14) = 0

Either x – 13 = 0 or x − 14 = 0

i.e., x = 13 or x = 14

If first number = 13, then

Other number = 27 − 13 = 14

If first number = 14, then

Other number = 27 − 14 = 13

Therefore, the numbers are 13 and 14.