Determine all possible pair(s) (x,y) of real numbers that satisfy this set of equations:

x+y = 23, and:

√(x² + 12*y) + √(y² + 12*x) = 33

*** For an extra challenge, solve this puzzle without the aid of an online equation solver or a computer program.

x+y=23 thus y=23-x, substituting this into the second we get
¡î(x©÷-12x+276) + ¡î(x©÷-34x+529) = 33 squaring both sides we get
2x©÷-46x+805+2¡î( (x©÷-12x+276)*(x©÷-34x+529) ) = 1089
(x©÷-12x+276)*(x©÷-34x+529) = (x©÷-23x-142)©÷
x^4-46x^3+1213x^2-15732x+146004=x^4-46x^3+245x^2+6532x+20164
thus
968x^2-2264x+125840=0
968(x^2-23x+130)=0
968(x-10)(x-13)=0
thus x=10 and y=13 or
x=13 and y=10
thus the only real solutions are
(10,13) and (13,10)

Posted by Daniel on 2010-09-25 13:36:26