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Question
Solve the following quadratic equations by factorization:
`(x-a)/(x-b)+(x-b)/(x-a)=a/b+b/a`
Solution
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.
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