Question

find 2 numbers sum is 27 product is 182

Answer 1

Let the numbers be a and b
Therefore,a+b=27
or,a=27-b..............(1)
ab=182
putting the value of a as 27-b from...(1) we get,
(27-b)b=182
or,27b-b²-182=0
or,b²-27b+182=0
or,b²-13b-14b+182=0
or,b(b-13)-14(b-13)=0
or,(b-14)(b-13)=0
Hence, either b-14=0 or b-13=0
or,either b=14 or b=13
Since a=27-b
a= either 27-14 or 27-13
or,a= either 13 or 14.
Therefore, either 
a and b are 13 and 14 or
a and b are 14 and 13 .
By both ways , we can see that the 2 numbers are 14 and 13.
Ans:The numbers are 14 and 13.

Answer 2

Let the two numbers be A and B.

         A + B = 27
          A B = 182 = 13*7*2*1

Let us find A and B by factors method.

A and B can be possibly from the list of the following :
         1, 2, 7, 13, 14, 26, 91, 182

Take the left most factor and right most factor and then check if sum is 27. Continue to the next factor from left.  So check 1 & 182 ; 2 & 91 ; 7 & 26 ; 

If you choose 13 and 14 as A and B, then A+B = 27 and AB = 182.

         So   A = 13   and  B = 14

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Another way:

        A+ B = 27         so           A = 27 - B

             A B = (27 - B)B = 27 B - B² = 182
      
                B² - 27 B + 182 = 0
                B = [ 27 + - √ ( 27² - 4*182)  ] / 2 =  (27+1) / 2  or  (27-1)/2

                     B = 14  or 13

           So  A = 27 - B = 13  or  14.