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 ) * * *