√(x-2) + √(x+3) = √(6x-11)
(√(x-2) + √(x+3))² = (√(6x-11))²
x-2 + 2√((x-2)(x+3)) + x+3 = 6x-11
2x+1+2√(x²-2x+3x-6) =6x-11
2√(x²+x-6) = 6x-2x-11-1
2√(x²+x-6) = 4x-12
2√(x²+x-6) = 2(2x-6)
√(x²+x-6)=2x-6
x²+x-6 =(2x-6)²
x²+x-6=4x²-24x+36
x²-4x²+x+24x-6-36=0
-3x²+25x-42=0
3x²-25x+42=0
D=25² -4*3*42=625-504=121
x₁= 25-11 = 14/6 = 7/3
6
x₂ = 25+11 =36/6 =6
6
Проверка корней:
х₁ = 7/3 √(7/3 -2) + √(7/3 +3) = √(6*(7/3) -11)
√(1/3) + √(16/3) = √3
1 + 4 = √3
√3 √3
5 = √3
√3
5√3 ≠ √3
3
х₁ = 7/3 - не корень уравнения.
х₂=6 √(6-2) + √(6+3) = √(6*6-11)
√4 + √9 = √25
2+3 =5
5=5
х₂ = 6
Ответ: 6.