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

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Question 1104242: Find the product (xy) if x+y+ = 20 and x-y+=12.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52814)   (Show Source): You can put this solution on YOUR website!
.
It is assumed in the given equations  that   >= 0  and   >= 0.


1.  Let u = .  Then

     = 20,   or

     = 0.

    Factor left side

    (u+5)*(u-4) = 0.

    As  just noticed above,  assumed to be >= 0.
    Therefore, only positive root  u = 4  works.

    Thus,   = 4.   Then  x + y = 4^2 = 16.



2.  Similarly, Let v = .  Then

     = 12,   or

     = 0.

    Factor left side

    (v+4)*(v-3) = 0.

    As  just noticed above,  assumed to be >= 0.
    Therefore, only positive root  v = 3  works.

    Thus,   = 3.   Then  x - y = 3^2 = 9.

    

3.  Thus we have two equations

    x + y = 16,
    x - y =  9,


    which implies  x= 12.5,  y= 3.5.



Answer.  The solution is  x= 12.5,  y= 3.5.  The product xy = 43.75.

Solved.


Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Let

Then the first equation is




Then or .

Since the square root has to be positive, , which means (1).

Similarly, let ; then the second equation is





Then or .

Then, as before, , which means (2).

Solving (1) and (2) together gives us x = 25/2, y = 7/2.

Then the product we are looking for is
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