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

Question

Since it is not in the form of quadratic equation , you have to make it into such an equation.

Answer 1

x-a)/(x-b) + (x-b)/(x-a) = a/b +b/a
multiply x-a
(x-a)^2/(x-b) + (x-b) = (a/b +b/a)(x-a)
multiply x-b
(x-a)^2 + (x-b)^2 =(a/b +b/a)(x-a)(x-b)
(x-a)^2 - (a/b +b/a)(x-a)(x-b) + (x-b)^2 = 0

[(x-a)-(a/b)(x-b)] [(x-a)-(b/a)(x-b)] = 0

So, (x-a)-(a/b)(x-b)=0
b(x-a)-a(x-b)=0
(b-a)x=0
x=0 where a<>b

So, (x-a)-(b/a)(x-b)=0
a(x-a)-b(x-b)=0
(a-b)x=a^2-b^2
(a-b)x=(a+b)(a-b)
x = a+b
where a<>b

Answer 2

(x - a)/(x - b) + (x - b)/(x - a) = a/b + b/a

Your first step is to multiply by the lowest common denominator of
(x - b) , (x - a) , b, and a. The LCD is (x - b)(x - a)(b)(a), and multiplying both sides by this should eliminate all fractions.

(x - a)(x - a)(b)(a) + (x - b)(x - b)(b)(a) = a(x - b)(x - a)(a) +
b(x - b)(x - a)(b)

First, let's avoid confusion by moving all constants to the front.

ab(x - a)^2 +ab(x - b)^2 = a^2(x - b)(x - a) + b^2(x - b)(x - a)

All we can do at this point is expand everything.

ab(x^2 - 2ax + a^2) + ab(x^2 - 2bx + b^2) = a^2(x^2 - ax - bx + ab) + b^2(x^2 - ax - bx + ab)

And expanding some more.

abx^2 - 2a^2 bx + a^3 b + abx^2 - 2ab^2 x + ab^3 = a^2 x^2 - a^3 x
- a^2 bx + a^3 b + b^2 x^2 - ab^2 x - b^3 x + ab^3

Group like terms,

abx^2 + abx^2 - a^2 x^2 - b^2 x^2 - 2a^2 bx - 2ab^2 x + a^3 x +
a^2 bx + ab^2 x + b^3 x + a^3 b + ab^3 - a^3 b - ab^3 = 0

(ab + ab - a^2 - b^2)x^2 + (-2a^2 - 2ab^2 + a^3 + a^2 + ab^2 + b^3)x + a^3 b + ab^3 - a^3 b - ab^3 = 0

This question's way too long. I'm stopping here.