Question

Solve the following quadratic equations by factorization:

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

Answer 1

We have been given

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

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

2abx² - 2(ab)(a + b)x + ab(a² + b²) = (a² + b²)x² - (a + b)(a² + b²)x + ab(a² + b²)

(a - b)²x² - (a + b)(a - b)²x = 0

x(a - b)²(x - (a + b)) = 0

Therefore,

x(a - b)² = 0

x = 0

or,

x - (a + b) = 0

x = a + b

Hence, x = 0 or x = a + b.