Find`sqrt(x^2 +2x+(y-1)^2 -2xy)` ` `

Please state your workings clearly.

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First evaluate the expression under the square root sign:

`x^2 + 2x + (y-1)^2 - 2xy = x^2 + 2x + y^2 - 2y + 1 - 2xy`

`= x^2 + y^2 -2xy + 2(x-y) + 1`

Now

`(x-y)^2 = x^2 + y^2 - 2xy`

So that

`[(x-y)+1][(x-y)+1] = (x-y)^2 + 2(x-y) + 1`

`= (x^2 + y^2 - 2xy) + 2(x-y) + 1`

which is the expression we require.

Hence the expression under the square root sign simplifies to

`[(x-y)+1][(x-y)+1] = [(x-y)+1]^2`

Thus, square rooting this the result is

`(x-y) + 1`

**answer**

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