Question

Solve for x
1/(x-1)(x-2)+1/(x-2)(x-3)+1/(x-3)(x-4)=1/6
When x is not equal to 1,2,3,4.

Answer 1

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1 + 1 = 2

--------------- ---------------- ----

(x - 1)(x - 2) (x - 2)(x - 3) 3

1 [ 1 + 1 ] = 2 -------------[Taken 1 / (x - 2) Common ]

------- --------- --------- ----

(x - 2) (x - 1) (x - 3) 3

1 [x - 3 + x - 1] = 2

------- ------------------ ----

(x - 2) (x - 1)(x - 3) 3

1 [ 2x - 4 ] = 2

------- ------------- ----

(x - 2) x² - 4x + 3 3

2x - 4 = 2

---------------------------- -----

x³ - 4x² - 2x² + 3x - 6 3

2x - 4 = 2

---------------------- -----

x³ - 6x² + 3x - 6 3

(2x - 4)3 = 2(x³ - 6x² + 3x - 6)

6x -12 = 2x³ -12x² +6x -12--[6x - 12 gets cancelled on both the sides]

2x³ - 12x² = 0

2x²( x - 6 ) = 0

2x² = 0 OR x - 6 = 0

x = 0 OR x = 6