Odpowiedź:
1) x² - 6x + 5 ≥ 0
Δ = b² - 4ac = 6² - 4(1)(5) = 36 - 20 = 16
[tex]\sqrt{delta}[/tex] = 4
x1 = [tex]\frac{-b-\sqrt{delta} }{2a}[/tex] = [tex]\frac{6-4}{2}[/tex] = 1
x2 = [tex]\frac{-b+\sqrt{delta} }{2a}[/tex] = [tex]\frac{6+4}{2}[/tex] = 5
x∈(-∞, 1> ∪ <5, +∞)
2) -2x² + 3x + 1 < 0
Δ = 3² - 4(-2)(1) = 9 - 8 = 1
[tex]\sqrt{delta}[/tex] = 1
x1 = [tex]\frac{-b-\sqrt{delta} }{2a}[/tex] = [tex]\frac{-3-1}{-4}[/tex] = 1
x2 = [tex]\frac{-b+\sqrt{delta} }{2a}[/tex] = [tex]\frac{-3+1}{-4}[/tex] = [tex]\frac{1}{2}[/tex]
x∈(-∞, [tex]\frac{1}{2}[/tex]) ∪ (1, +∞)
3) -2x² + 6x - [tex]\frac{9}{2}[/tex] ≤ 0 |×2
-4x² + 12x - 9 ≤ 0
Δ = 12² - 4(-4)(-9) = 144 - 144 = 0
x = [tex]\frac{-b}{2a}[/tex] = [tex]\frac{-12}{-8}[/tex] = [tex]\frac{3}{2}[/tex]
x∈R
4) 6x² - x > 12
6x² - x - 12 > 0
Δ = 1 - 4(6)(-12) = 1 + 288 = 289
[tex]\sqrt{delta}[/tex] = 17
x1 = [tex]\frac{-b-\sqrt{delta} }{2a}[/tex] = [tex]\frac{1-17}{12}[/tex] = -[tex]\frac{4}{3}[/tex]
x2 = [tex]\frac{-b+\sqrt{delta} }{2a}[/tex] = [tex]\frac{1+17}{12}[/tex] = [tex]\frac{3}{2}[/tex]
x∈(-∞, -[tex]\frac{4}{3}[/tex]) ∪ ([tex]\frac{3}{2}[/tex], +∞)
5) x² - x ≥ 6
x² - x - 6 ≥ 0
Δ = 1² - 4(1)(-6) = 1 + 24 = 25
[tex]\sqrt{delta}[/tex] = 5
x1 = [tex]\frac{-b-\sqrt{delta} }{2a}[/tex] = [tex]\frac{1-5}{2}[/tex] = -2
x2 = [tex]\frac{-b+\sqrt{delta} }{2a}[/tex] = [tex]\frac{1+5}{2}[/tex] = 3
x∈(-∞, -2> ∪ <3, +∞)
