You are given


    a   + b   = 25    (1)

    a^2 + b^2 = 337.  (2)


Square equation (1). Keep equation (2) as is.  You will get


    a^2  + 2ab + b^2 = 625    (1')

    a^2 +        b^2 = 337.   (2')


From equation (1'), subtract equation (2').  You will get


          2ab        = 625 - 337 = 288,   or

           ab                    = 144.


Thus you need to find "a" and "b" in the way that


    a + b = 25,

    ab    = 144.


You can reduce it to the quadratic equation  a*(25-a) = 144 and solve it using Algebra.


But you can (alternatively) to guess the solution in 8 seconds.


ANSWER.  The solution is two pairs  (a,b) = (16,9)  and  (a,b) = (9,16).