Упростите [tex]\sqrt{x+2\sqrt{x-1} } + \sqrt{x-2\sqrt{x-1} } (1\leq x\leq 2)[/tex]

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task/29839281 Упростите  √[ x +2√(x-1) ] + √[ x -2√(x-1) ]    ( 1 ≤ x  ≤2)

решение √[ x +2√(x-1) ] +√[ x - 2√(x-1) ] =√[ x -1 +2√(x-1) +1 ] +√[ x -1 -2√(x-1) +1 ] =

√[√(x -1) + 1]² + √[ √(x -1) - 1] ²  = |√(x -1) + 1 | + | √(x -1) - 1 |  = √(x -1) + 1 -(√(x -1) - 1) =2.

*  *  *  1 ≤  x  ≤ 2  ⇔ 0 ≤ x -1  ≤  1 ⇔ 0  ≤ √(x -1 )  ≤ 1 ⇔-1 ≤ √(x -1 ) - 1 ≤  0

⇒ |√(x -1 ) - 1 |  = - (√(x -1 ) - 1 )  * * *