x-a/x-b + x-b/x-a = a/b + b/a (sove for x using quadratic formula)?

Question

ONLY use quadratic formula . D = b^2 - 4ac

Answer 1

x-a/x-b + x-b/x-a = a/b + b/a

Multiplying across by ab(x-a)(x-b)...

ab(x-a)² +ab(x-b)² = (a²+b²)(x-a)(x-b)

ab(x²-2ax+a² +x²-2bx+b²) = (a²+b²)(x²-(a+b)x+ab)
ab(2x²-2(a+b)x+a²+b²) = (a²+b²)(x²-(a+b)x+ab)

Making the algebraic substitution e=(a+b), f=ab
to make it more tractable, and noting (a²+b²)=e²-f:
f(2x²-2ex+e²-f) = (e²-f)(x²-ex+f)
2fx² -2efx +f(e²-f) = (e²-f)x² - e(e²-f)x +f(e²-f)
2fx² -2efx = (e²-f)x² - e(e²-f)x

Dividing across by x = 0 which is one solution (as Raichu comments)
(2f + f-e²)x = 2ef - e(e²-f)
x = e(f-e²) / (3f-e²)
x = e(e²-f) / (e²-3f)


Substituting back for e=(a+b), f=ab
x = (a+b)((a+b)²-ab) / ((a+b)²-3ab)
x = (a+b)(a²+ab+b²) / (a²-ab+b²)

Looks as good as you can get it.

If you try the formula you get painful experiences like below:

________________________________________________

So applying:
x = [ -B ± √(B² - 4AC) ] / 2A
with
A = 2
B = -2(a+b)
C = (a²+b²)(ab-1)/ab...

x = [ 2(a+b) ± √(4(a+b)² - 8(a²+b²)(ab-1)/ab)... ] / 4

Crunching out the determinant D, or D² for convenience
D = √(4(a+b)² - 8(a²+b²)(ab-1)/ab)
D² = 4(a²+2ab+b²) - 8(a²+b²)(ab-1)/ab
= [ 4a³b+8a²b²+4ab³ -8a³b -8ab³ +8a² +8b² ] /ab
= 4[ -a³b+2a²b²-ab³ +2a² +2b² ] /ab

If you make some algebraic substitution like u=(a+b), v=ab
then maybe you get something more tractable:
A = 2
B = -2u
C = (u²-v)(v-1)/v

Answer 2

I presume you mean (x-a)/(x-b) + (x-b)/(x-a) = a/b + b/a.

Multiply both sides by (x-a)(x-b) and you get
(x-a)^2 + (x-b)^2 = (a/b + b/a)(x-a)(x-b).

Rewrite as
(x-a)^2 + (x-b)^2 - (a/b + b/a)(x-a)(x-b) = 0.

Expand the brackets, collect like terms and simplify. You'll end up with a quadratic of the form

Ax^2 + Bx + C = 0

which is solved by the quadratic formula

(-B +or- sqrt(D)) / 2A where D= B^2 - 4AC.

In fact, I tried solving it and it simplifies so that you don't even need the formula. You can see it because in fact x=0 is a solution which makes C=0, so the other solution is x=-B/A.