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Question
Solve the following equation: `"a"/("x" - "a") + "b"/("x" - "b") = (2"c")/("x" - "c")`
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Solution
`"a"/("x" - "a") + "b"/("x" - "b") = (2"c")/("x" - "c")`
`("a"("x" - "b") + "b"("x" - "a"))/(("x" - "a")("x" - "b")) = (2"c")/("x" - "c")`
`("ax - ab + bx - ab")/("x"^2 - "ax" - "bx" + "ab") = (2"c")/("x" - "c")`
`(("a + b")"x" - 2 "ab")/("x"^2 - ("a + b")"x" + "ab") = (2"c")/("x" - "c")`
{(a+ b) x- 2ab} (x- c) = 2c {x2 - (a + b) x + ab}
(a+ b )x2 - 2abx - c (a+ b )x + 2abc = 2cx2 - 2c( a+ b )x + 2abc
(a+ b )x2 - [2ab + c( a+ b )]x + 2abc = 2cx2 - 2c( a+ b )x + 2abc
(a+ b-2c)x2 = (2ab + ac+ bc- 2ca - 2bc)x
(a+ b-2c )x2 = (2ab - ac - bc) x
x = 0 , x = `(2"ab" - "ac" - "bc")/("a" + "b" - 2"c")`
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