2014/059) find real solutions

a- b  + c - d = 0 .... (1)
ab = cd  ...(2)
a^2 - b^2 + c^2 - d^2 = - 24 ...(3)
a^2 + b^2 + c^2 + d^2 = 50 ...(4)

from (1)
a- b = d- c... (5)

square above and using (2)

(a-b)^2 + 4ab = (c-d)^2 + 4cd

or (a+b)^2 = (c+d)^2

hence a + b = c + d ... (6)
or a+ b = -c - d ..(7)
from (5) and (6) a = d and b= c but it is not possible as LHS of (3) is zero which is contradiction
from (5) and (7) a = -c and b = - d
so we get from (3) and (4)
a^2 - b^2 = - 12
a^2 + b^2 = 25

add above to get 2 a^2 = 13, subtract to get 2b^2 = 37

this gives 4 set of solutions

(a,b,c,d) = ((13/2)^(1/2), (37/2)^(1/2), -(13/2)^(1/2),-  (37/2)^(1/2))
or  ((13/2)^(1/2),- (37/2)^(1/2), -(13/2)^(1/2),  (37/2)^(1/2))
or (- (13/2)^(1/2), (37/2)^(1/2), (13/2)^(1/2),-  (37/2)^(1/2))
or (-(13/2)^(1/2),-  (37/2)^(1/2), (13/2)^(1/2), (37/2)^(1/2))