Question

Fill in the missing term in the equation. x/x^2-4x+4-x/x^2-3x+2=?/(x-2)^2(x-1)

Answer 1

The missing term is x.

Explanation:

Given expression is,

`(x)/(x^2-4x+4)-(x)/(x^2-3x+2)------(1)`

Since,
`x^2-4x+4=(x)^2-2* (2x)+(2)^2`

`\implies x^2-4x+4=(x-2)^2------(2)` ( (a-b)² = a² - 2ab + b² )

Now,
`x^2-3x+2=x^2-2x-x+2` ( By middle term splitting )

`=x(x-2)-1(x-2)`

`\implies x^2-3x+2=(x-1)(x-2)------(3)`

From equation (1), (2) and (3),

`(x)/(x^2-4x+4)-(x)/(x^2-3x+2)=(x)/((x-2)^2)-(x)/((x-2)(x-1))`

`=(x(x-1)-x(x-2))/((x-2)^2(x-1))`

`=(x(x-1-x+2))/((x-2)^2(x-1))`

`=(x(1))/((x-2)^2(x-1))`

`=(x)/((x-2)^2(x-1))`

Hence, the missing term in the equation is x.

Answer 2

x x
----------------- - --------------------
x^2 -4x + 4 x^2-3x+2

x x
= ---------------- - --------------------
(x -2)^2 (x - 2)(x - 1)

x (x-1) - x(x +2)
= ----------------------------
(x -2)^2 (x - 1)

x^2 -x -x^2 - 2x
= --------------------------
(x -2)^2 (x - 1)

-3x
= -----------------------
(x -2)^2 (x - 1)

answer
missing term in the equation: -3x