Let A be the first number and B be the second number.

then A+B = 55 and A*B = 684

Solve for one variable

A = 55-B

Substitute it's value into the other equation so that you only have one variable in the other equation

(55-B)*B = 684

Set equal to zero so that quadratic formula can be used

-B^2 -55B -684 = 0

Use the quadratic formula The Quadratic Formula Explained to solve for B, and then plug that back into either equation to solve for A