Question

Answer ⬆️.

With appropriate explanation.

Answer 1

a/(x-b) + b/(x-a)= 2

a/(x-b)= 2 - b/(x-a)

a/(x-b)= 2x - 2a -b ) / (x-a)

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

ax +2ax +bx +2bx = b^2 + 2ab +a^2 +2x^2

3ax + 3bx = ( b+a)^2 + 2x^2

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

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

x( 2x - (a+b)) - ( b+a)( 2x - ( b+a)) = 0

x-( b+a)( 2x - (a+b)) = 0

x= b+a, (a+b)/2

Answer 2

[ a / ( x - b )] + [ b ( x - a ) ] = 2

[ a ( x - a ) + b ( x - b )] / [ ( x - a ) ( x - b )] = 2

[ ax - a^2 + bx - b^2] / [ x^2 - bx - ax + ab] = 2

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

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

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

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

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

0 = x [ 2x - ( a + b )] - ( a + b ) [ 2x - ( a + b )]

0 = [ x - ( a + b )] [ 2x - ( a + b )]

$<h4>x = ( a + b ), or ( a + b ) / 2$