If aligned & A+ B=m & A+ B=n aligned where m, n 0 , then what is (A+B) equal to?

Step-by-step solution

aligned & m n= A ~A + A ~B & + A ~B + B ~B & = 2 ~A 2 + ( A + B )+ 2 ~B 2 & = ( A + B )+ 1 2 2 ( ~A + B ) ( A - B ) & = ( A + B )+ ( A + B ) ( A - B ) & m n= ( A + B ) 1+ ( A - B ) ...(i) aligned Also, aligned m^2+n^2= & ^2 ~A + ^2 ~B & +2 ~A ~B + + & ^2 ~A + ^2 ~B & +2 ~A ~B aligned aligned & =2+2 ( ~A - B ) & m^2+n^2=2 1+ ( A - B ) & m^2+n^2 2 =1+ ( A - B )...(ii) aligned From (i) and (ii) ( A + B )= 2 m n m^2+n^2