Răspuns:
lg 2x - 6 lg x + 5 = 0
2 lg x - 3 lg x² + 5 = 0
Notam lg x cu t, astfel:
-3t² + 2t + 5 = 0
Δ = b² - 4ac = 4 + 60 = 64
x₁ = ( -b + √Δ ) / 2a = ( -2 + 8 ) / -6 = 6 / -6 = -1
x₂ = ( -b - √Δ ) / 2a = ( -2 - 8 ) / -6 = -10 / -6 = 10 / 6 = 5 / 3
t = lgx =>
1. lg x = -1 => x = 10⁻¹ = [tex]\frac{1}{10}[/tex]
2. lg x = 5 / 3 => x = [tex]10^{5/3}[/tex] = ∛10⁵
=> S = { [tex]\frac{1}{10}[/tex] , ∛10⁵ }
EDIT: lg²x - 6 lgx + 5 = 0
C.E: x > 0, x ≠ 1 => x ∈ ( 0, +∝ ) / { 1 }
Notam lgx cu t, astfel:
t² - 6t + 5 = 0
Δ = b² - 4ac = 36 - 20 = 16
x₁ = ( -b + √Δ ) / 2a = ( 6 + 4 ) / 2 = 10 / 2 = 5
x₂ = ( -b - √Δ ) / 2a = ( 6 - 4 ) / 2 = 2 / 2 = 1
1. lgx = 5 => x = 10⁵
2. lgx = 1 => x = 10
=> S = { 10, 10⁵ } ∈ ( 0, +∝ ) / { 1 }
