Вправа 301 - 400 » 357 (в-г)

в) ax/(a+x) – bx/(b-x) = (a • ab/(a+b))/(a+ ab/(a+b)) – (b • ab/(a+b))/(b- ab/(a+b)) = (a^2 b(a+b))/((a+b)(a^2+ ab+ab)) – (ab^2 (a+b))/((a+b)(ab+ b^2- ab)) = (a^2 b)/(a^2+ 2ab) – ab^2/b^2 = (a^2 b)/(a^2+ 2ab) – a = (a^2 b- a^3- 2a^2 b)/(a^2+ 2ab) = (a(ab- a^2- 2ab))/(a^2+ 2ab) = (〖-(a〗^2+ ab) •a)/((a^2+ 2ab)) = (〖-a〗^2 (a+b))/(a(a+2b)) = (-a(a+b))/(a+2b); г) ((x+2b)/(x-2b) + (x+2a)/(x-2a)) : x/2 = ((x+2b)(x-2a)+ (x+2a)(x-2b))/((x-2b)(x-2a)) • 2/x = (x^2- 2ax+2xb -4ab+x2-2bx+2ax-4ab)/((x-2b)(x-2a)) • 2/x = (〖2x〗^2- 8ab)/((x-2b)(x-2a)) • 2/x = (4(x^2- 4ab))/x(x-2b)(x-2a) . Якщо x = 4ab/(a+b), то (4 •((〖16a〗^2 b^2)/(a+b)^2 - 4ab))/(4ab/(a+b) •(4ab/(a+b)- 2b)(4ab/(a+b-2a))) = (16(〖4a〗^2 b^2- a^3 b- 〖2a〗^2 b^2- ab^3 )/(a+b)^2 )/(4ab/(a+b) •(〖4ab-2ab-2b〗^2/(a+b))((4ab- 〖2a〗^2- 2ab)/(a+b)) ) = (〖16(4a〗^2 b^2- a^3 b- 〖2a〗^2 b^2- 〖ab〗^3)(〖a+b)〗^3)/((〖a+b)〗^2 •4ab(2ab- 〖2b〗^2)(2ab- 〖2a〗^2)) = (16(b-a)(a^2 b- 〖ab〗^2 )(a+b))/(4ab •4b •a(a-b)(b-a)) = (ab(b-a)(a-b)(a+b))/(ab •ab(a-b)(b-a)) = (a+b)/ab.