Question

Solve the following quadratic equation by factorization.
$\frac{x-a}{x-b}+\frac{x-b}{x-a}=\frac{a}{b}+\frac{b}{a}$

Answer 1

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

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

= (a²+b²-2ab)x-(a+b)(a²+b²-2ab)x=0

= (a-b)²x²-(a+b)(a+b)²x²=0

= x(a-b)²(x-(a+b))=0

= x(x-(a+b))=0

Either x= 0

Or, (x-(a+b))=0

Therefore x= a+b

The roots of the above mentioned quadratic equation are 0 and a+b respectively.