Odpowiedź:
a)
(x² - 4x - 5)/(x³ - x) = 0
założenie:
x³ - x ≠ 0
x(x² - 1) ≠ 0
x(x - 1)(x + 1) ≠ 0
x ≠ 0 ∧ x ≠ 1 ∧ x ≠ - 1
D: x ∈ R - {- 1 , 0 , 1 }
x² - 4x - 5 = 0
a = 1 , b = - 4 , c = - 5
Δ = b² - 4ac = (- 4)² - 4 * 1 * (- 5) = 16 + 201 = 36
√Δ = √36 = 6
x₁ = ( - b - √Δ)/2a = (4 - 6)/2 = - 2/2 = - 1
x₂ = (- b + √Δ)/2a = ( 4 + 6)/2 = 10/2 = 5
b)
(x + 7)/(x - 7) = 0
założenie :
x - 7 ≠ 0
x ≠ 7
D: x ∈ R - {7}
x + 7 = 0
x = - 7
c)
1/x + 1 = 4/(x + 1)
założenie:
x ≠ 0 ∧ x + 1 ≠ 0
x ≠ 0 ∧ x ≠ - 1
D: x ∈ R - {- 1 , 0 }
(1 + x)/x = 4/(x + 1)
4x = (x + 1)(x + 1)
4x = (x + 1)² = x² + 2x + 1
x² + 2x + 1 - 4x = 0
x² - 2x + 1 = 0
a = 1 , b = - 2 , c = 1
Δ = b² - 4ac = (- 2)² - 4 * 1 * 1 = 4 - 4 = 0
x₁ = x₂ = - b/2a = 2/2 = 1
d)
[x(x₂ - x - 6)]/(x² - 4)
założenie:
x² - 4 ≠ 0
(x - 2)(x + 2) ≠ 0
x - 2 ≠ 0 ∧ x + 2 ≠ 0
x ≠ 2 ∧ x ≠ - 2
D: x ∈ R - {- 2 , 2 }
x(x² - x - 6) = 0
x = 0 ∨ x² - x - 6 = 0
x² - x - 6 = 0
a = 1 , b = - 1 , c = - 6
Δ = b² - 4ac = (- 1)² - 4 * 1 * (- 6) = 1 + 24 = 25
√Δ = √25 = 5
x₁ = ( - b - √Δ)/2a = (1 - 5)/2 = - 4/2 = - 2
x₂ = ( - b + √Δ)/2a = (1 + 5)/2 = 6/2 = 3
x = 0 ∨ x = - 2 ∨ x = 3
e)
(x - 1)/(x + 2) = 2x/(x + 2)
założenie:
x + 2 ≠ 0
x ≠ - 2
D: x ∈ R - { - 2}
(x - 1)(x + 2) = 2x(x + 2)
x² - x + 2x - 2 = 2x² + 4x
x² + x - 2 = 2x² + 4x
x² - 2x² + x - 4x - 2 = 0
- x² - 3x - 2 = 0
a = - 1 , b = - 3 , c = - 2
Δ = b² - 4ac = (- 3)² - 4 * (- 1) * ( - 2) = 9 - 8 = 1
√Δ = √1 = 1
x₁ = ( - b - √Δ)/2a = ( 3 - 1)/(- 2) = 2/(- 2) = - 2/2 = - 1
x₂ = ( - b + √Δ)/2a = ( 3 + 1)/(- 2) = 4/(- 2) = - 4/2 = - 2
