Question

Prove that, a^2/x-b+b^2/x-a=a+b

Answer 1

Given : a²/x-b+b²/x-a=a+b

To Find : prove that x = a + b

Solution:

a²/x-b+b²/x-a=a+b

Multiplying both sides by (x - a) and (x - b)

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

=> x(a² + b² )  - (a³ + b³ ) = (a + b) ( x²  -(a + b)x  + ab)

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

=>  (a + b) x²  - x (a + b)²  -x (a² + b² )  +  (a + b) ab + (a  + b  )(a² + b² - ab)  = 0

=>  (a + b) x²  - x (a + b)²  -x (a² + b² )  +  (a + b)  (ab + a² + b² - ab)  = 0

=>  (a + b) x²  - x (a + b)²  -x (a² + b² )  +  (a + b)  ( a² + b² )  = 0

=>  (a + b)x{x   -  (a + b) }   -   (a² + b² )(x  - (a  + b)  ) = 0

=> {x   -  (a + b) } ((a + b)x -  (a² + b² )) = 0

x   -  (a + b) = 0

=> x = a+b

QED

or (a + b)x -  (a² + b² ) = 0  => x =  (a² + b² )/ (a + b)

x = a+b     or  x =  (a² + b² )/ (a + b)

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