Question

Solve for x

:`1/((x-1)(x-2))+1/((x-2)(x-3))=2/3` , x ≠ 1,2,3

Answer 1

`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