this may require some calculator help?

Question

Let .5(a+1/a) = -( 2+x) , .5(b+1/b) = -2. I need to to express the following in terms of 'x'. " .5[ (a+b)/(1+ab) + (1+ab)/(a+b) ] "

you are a super genius if you get this.

Answer 1

(a+1/a) = -(4+2x) = (a² +1)/a
(b+1/b) = -4 = (b²+1)/b

define h = a/(a²+1) + b/(b²+1)
h = [ab² + a + a²b + b] / [a²b² + a² + b² + 1]
......... = -1/(4+2x) - 1/4 = -(x+4)/(4x+8)

Now,
[ (a+b)/(1+ab) + (1+ab)/(a+b) ]
= [(a+b)² + (1+ab)²] / [(1+ab)(a+b)]
= [a²+2ab+b²+1+2ab+a²b²] / [ab² + a + a²b + b]
= [a²b² + a² + b² + 1]/ [ab² + a + a²b + b] +
........ 4ab/ [ab² + a + a²b + b]
= 1/h + 4/[a+1/a + b+1/b]
= -(4x+8)/(x+4) + 4/[-(4+2x)-4]
= -(4x+8)/(x+4) - 2/(x+4)
= -(4x+10)/(x+4)

The final answer is:
-(2x+5)/(x+4) .............. forgot the 0.5 in front.


d:

Answer 2

((a + 1)/a) = - 2( 2 + x)
((a + 1)/a) = - 4 - 2x
a + 1 = - 4a - 2ax
5a + 2ax = - 1
a = - 1/(2x + 5)

(b + 1)/b = - 4
b + 1 = - 4b
5b = - 1
b = - 1/5

.5[ (a + b)/(1 + ab) + (1 + ab)/(a + b) ] =

(1/2)[ ((a + b)^2 + (1 + ab)^2)/(a + b)(1 + ab) ] =

(1/2)[ ((- 1/(2x + 5) - 1/5)^2 + (1 + (- 1/(2x + 5))(- 1/5))^2)/(- 1/(2x + 5) - 1/5)(1 + (- 1/(2x + 5))(- 1/5)) ] =

(1/2)[ ((- 5 - 2x + 5)/(5(2x + 5)))^2 + (1 + (1/(5(2x + 5))))^2/(- 5 - 2x + 5)/(5(2x + 5)))(1 + (1/(5(2x + 5))) ] =

(1/2)[ (- 2x/(5(2x + 5)))^2 + (1 + (1/(5(2x + 5))))^2)/(- 2x/(5(2x + 5)))(1 + (1/(5(2x + 5))) ] =

(1/2)[ (4x^2 / (25(2x + 5)^2) + (1 + 2 / (5(2x + 5)) + 1 / (25(2x + 5)^2)) / ((- 2x / (5(2x + 5))) + (- 2x / (5(2x + 5)))(1 / (5(2x + 5))) ] =

(1/2)[ (4x^2 + ((25(2x + 5))^2 + 2 (5(2x + 5)) + 1) / ((- 2x(5(2x + 5)) - 2x) ] =

(1/2)[ (4x^2 + 25(4x^2 + 20x + 25) + 20x + 50 + 1) / ((- 2x(10x + 25 + 1) ] =

- [ (104x^2 + 520x + 676) / (40x^2 + 104x) ] =

- (26x^2 + 130x + 169) / (x(10x + 26))