Question

Exercitiile 9, 10 va rog
Dau coroana

Answer 1

9) Δ = 4(m - 2)² - 4(m - 1)(m - 3) = 4m² -16m + 16 -4(m² -m - 3m + 3) =
= 4m² - 16m + 16 - 4m² +16m - 12 = 16 - 12 = 4
x1,2 = [-2(m - 2) +/- 2] / 2
x1 = (-2m + 4 - 2) / 2 = (-2m + 2)/2 = 1 - m
x2 = (-2m + 4 + 2) / 2 = 3 - m
a) x1 > 0 si x2 > 0
1 - m > 0, m < 1
3 - m > 0, m < 3
m ∈ (-∞, 1)

x1 < 0, x2 < 0
m > 1
m > 3
m ∈ (3, ∞)

b) x1 > 0 si x2 < 0
m < 1 si m > 3 nu se poate

x1< 0 si x2 > 0
m > 1 si m <3 deci m ∈ (1, 3)

10)
a) 6x² - 7x + 2 = 0
Δ = 49 - 48 = 1
x1,2 = (7 +/- 1)/12
x1= 8/12 = 2/3
x2 = 1/2
6x² - 7x + 2= (x - 2/3)(x - 1/2)

b) x² - x - 1= 0
Δ = 1 + 4 = 5
x1 = (1 + √5)/2
x2 = (1 - √5)/2
x² - x - 1= [x - (1 + √5)/2] [ x -  (1 - √5)/2]

c) 2x² -7mx + 6m² = 0
Δ = 49m² - 48m² = m²
x1 = (7m + |m|)/4
x2 = (7m - |m|)/4
Pentru m < 0, |m| = - m
x1 = 6m/4 = 3m/2
x2 = 8m/4 = 2m
Pentru m ≥ 0, |m| = m
x1 = 2m
x2 = 3m/2
2x² - 7mx + 6m² = (x - 2m) ( x - 3m/2)