Question

Rozwiąż równanie:
a) 2x-1/5x+4=2,
b) 3+2x/x-5=4x/2x-1
c)2/2x+3 - x-1/x-3=0
d)x/6-2x=2/x-3

Answer 1

A) 2x-1/5x+4=2 /* (5x + 4)
2x - 1 = 2 * (5x + 4)
2x - 1 = 10 x + 8
2x - 10 x = 8 +1
-8x = 9 /: (-8)
x = -9/8
x = - 1 1/8

b) 3+2x/x-5=4x/2x-1 /* (x - 5) (2x - 1)
(3 + 2x) * (2x -1) = 4x (x - 5)
6x - 3 + 4x - 2x = 4x² - 5x
- 4x² + 5x + 6x - 3 + 4x - 2x = 0
- 4x² + 13x - 3 = 0
Δ = b² - 4ac = 13² - 4* (-4) * (-3) = 169 - 48 = 121
Δ = √121
Δ = 11
x₁ = (- b - √Δ) : 2a = (- 13 - 11) : (-8) = -24 : -8 = 3
x₂ = (- b + √Δ) : 2a = (- 13 + 11) : (-8) = -2 : -8 = 1/4

c)2/2x+3 - x-1/x-3 = 0 /* (2x + 3) (x - 3)
2 * (x - 3) - (x - 1) * (2x + 3) = 0
2x - 6 - (2x² + 3x - 2x -3) = 0
2x - 6 - 2x² - 3x + 2x + 3 = 0
- 2x² - x - 3 = 0
Δ = b² - 4ac = (-1)² - 4* (-2) * (-3) = 1 + 24 = 25
Δ = √25
Δ = 5
x₁ = (- b - √Δ) : 2a = (1 - 5) : (-4) = -4 : -4 = 1
x₂ = (- b + √Δ) : 2a = (1 + 5) : (-4) = 6 : -4 = - 1 1/2

d)x/6-2x=2/x-3 / (6 - 2x) * (x -3)
x * (x - 3 ) = 2 * (6 - 2x)
x² - 3x = 12 - 4x
x² - 3x - 12 + 4x = 0
x² + x - 12 = 0
Δ = b² - 4ac = (1)² - 4* (1) * (-12) = 1 + 48 = 49
Δ = √249
Δ = 7
x₁ = (- b - √Δ) : 2a = (- 1 - 7) : (2) = -8 : 2 = - 4
x₂ = (- b + √Δ) : 2a = (-1 + 7) : (2) = 6 : 2 = 3