154371

for problem e)
Let A^2=a and
1 B^2=b and A^2=a
2 We now have
3 (AB)/(A+B)=(+or-)1
4 AB=A+B or AB=-A-B
5 Squaring both sides for both equations above we have the same answer ( this 6 is why line 3 is equal to +or- 1):
7 (A^2)(B^2)=(A^2)+2AB+(B^2)
8 Subtracting (A^2)(B^2) from both sides,
9 (A^2)-(A^2)(B^2)+2AB+(B^2)=0
10 Grouping in terms of A, we have
11 (1-(B^2))A^2+2BA+B^2=0
12 Solving for A in terms of B, we have
13 A=(-2B(+or-)sqrt(4(B^2)-4(B^2)(1-(B^2))))/(2(1-(B^2)))
14 Simplifying, we get
15 A=(-2B(+or-)2(B^2))/(2(1-(B^2))
16 A=(-B(+or-)(B^2))/(1-(B^2))
17 Factorising and simplifying gets us to our solutions for A in terms of B
18 A=-(B/(B+1))
19 or
20 A=B/(B-1)
21 A and B can be negative or positive, for A^2(which is a) is always positive 22and B^2(which is b) is always positive
23 Substituting -(B/(B+1)) for A into (AB)/(A+B) and simplifying, we get
24 (-(B^2)/(B+1))/((B^2)/(B+1))=-1
25 Substituting B/(B-1) for A into (AB)/(A+B) and simplifying, we get
26 ((B^2)/(B-1))/((B^2)/(B-1))=1
27 lines 23-26 were checking processes