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Question
Solve for x
:`1/((x-1)(x-2))+1/((x-2)(x-3))=2/3` , x ≠ 1,2,3
Solution
`1/((x-1)(x-2))+1/((x-2)(x-3))=2/3`
`:.((x-3)+(x-1))/((x-1)(x-2)(x-3))=2/3`
`:.(x-3+x-1)/((x^2-2x-x+2)(x-3))=2/3`
`:.(2x-4)/((x^2-3x+2)(x-3))=2/3`
`:.(2x-4)/(x^3-3x^2-3x^2+9x+2x-6)=2/3`
`:.(2x-4)/(x^3-6x^2+11x-6)=2/3`
∴ 6x - 12 = 2x3 -12x2 + 22x - 12
∴ 2x3 - 12x2 + 16x = 0
∴ 2x(x2-6x+8) = 0
∴ (x2-6x+8) = 0
∴ x2 - 4x - 2x + 8 = 0
∴ x(x- 4)- 2(x - 4)= 0
∴ (x-4)(x-2)=0
∴ x-4=0 or x-2 = 0
∴ x = 4 or x = 2
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