SOLUTION: Find the product(xy) if {{{ x+y+ sqrt( x+y ) = 20 }}} and {{{ x-y+ sqrt( x-y ) = 12 }}}

Let   = p.     (1)

Then first equation takes the form

    p^2 + p - 20 = 0,  or, factoring,  (p+5)*(p-4) = 0.     (2)

Its roots are p= -5 and p= 4.


Due to definition of p  (1), we suppose, by default, that "p" is positive; so we choose p= 4.

Hence,  x + y = 4^2 = 16.     (3)



Let   = q.     (4)

Then second equation takes the form

    q^2 + q - 12 = 0,  or, factoring,  (q+4)*(q-3) = 0.     (5)

Its roots are q= -4 and q= 3.


Due to definition of q  (4), we suppose, by default, that "q" is positive; so we choose q= 3.

Hence,  x - y = 3^2 = 9.     (6)



Thus we have these two equations

    x + y = 16     (3)

    x - y =  9     (6)


Solving by elimination, we get  x =  =  = 12.5;  y = 16-x = 16-12.5 = 3.5.


THEREFORE,  x*y = 12.5*3.5 = 43.75.      ANSWER