Решение
1) √(5 + x²) + √(5 - x²) = 4
ОДЗ: 5 + x² ≥ 0, 5 - x² ≥ 0,
[√(5 + x²)]² = [4 - √(5 - x²)]²
5 + x² = 16 - 8*√(5 - x²) + 5 - x²
2x² - 16 = - 8*√(5 - x²)
16 - 2x² = 8*√(5 - x²) делим обе части уравнения на 2
8 - x² = 4*√(5 - x²)
[8 - x²]² = [4*√(5 - x²)]²
64 - 16x² + x⁴ = 4*(5 - x²)
64 - 16x² + x⁴ = 20 - 4x²
x⁴ -12x² + 44 = 0
x⁴ - 15x² + 44 = 0
x² = y
y² - 15y + 44 = 0
D = 225 - 4*1*44 = 225 - 176 = 49
y₁ = (15 - 7)/2
y₁ = 4
y₂ = (15 + 7)/2
y₂ = 11
x² = 4
x₁ = - √4 = 2
x₂ = √4 = 2
x² = 11
x₃ = - √11
x₄ = √11
Проверка:
х = - 4
√(5 + (-2)²) + √(5 - (-2)²) = 3 + 1 = 4
4 = 4 Верно
x = 4
√(5 + 2²) + √(5 - 2²) = 3 + 1 = 4
4 = 4 Верно
x = - √11
√(5 + (-√11)²) + √(5 - (-√11)²) =4 + √(-6) не имеет смысла
x = √11
√(5 + (√11)²) + √(5 - (√11)²) =4 + √(-6) не имеет смысла
Ответ: x = - 2; x = 2
2) √[x - 4√(x + 4)] = 3
ОДЗ:
[√(x - 4√(x + 4)]² = 3²
x - 4√(x + 4) = 9
4√(x + 4) = x - 9
[4√(x + 4)]² = (x - 9)²
16*(x + 4) = x² - 18x + 81
x² - 18x + 81 - 16x - 64 = 0
x² - 34x + 17 = 0
D = 1156 - 4*1*17 = 1088
x₁ = (34 - 8√17)/2
x₁ = 17 - 4√17
x₂ = (34 + 8√17)/2
x₂ = 17 + 4√17