Odpowiedź:
Szczegółowe wyjaśnienie:
Zad. 1
- 3x + 2y - 1 = 0
2y = 3x + 1/:2
y = 1,5x + 0,5
a₁ * a₂ = - 1
1,5 * a₂ = - 1/: 1,5
a₂ = [tex]-\frac{10}{15}[/tex] = [tex]-\frac{2}{3}[/tex]
[tex]-\frac{2}{3}[/tex] * 1 + b = - 3
[tex]-\frac{2}{3}[/tex] + b = - 3
b = -3 + [tex]\frac{2}{3}[/tex] = [tex]-2\frac{1}{3}[/tex]
równanie prostej: y = [tex]-\frac{2}{3}x - 2\frac{1}{3}[/tex]
Zad. 2
- 3x(x -4) ≥ x² -16
- 3x² - 12x ≥ x² - 16
- 3x² - 12x - x² + 16 ≥ 0
-4x² - 12x + 16 ≥ 0
Δ = (-12)² - 4 * (-4) * 16 = 144 + 256 = 400
√Δ = 20
x₁ = [tex]\frac{12 - 20}{2 * (-4)} = \frac{-8}{-8}[/tex] = 1
x₂ = [tex]\frac{12 + 20}{2 * (-4)} = \frac{32}{-8} = -4[/tex]
x∈ <-4, 1>
Zad. 3
5x(x² + 9)(4-3x)(7-x²) = 0
5x = 0/:5
x₁ =0
x² + 9 = 0
Δ = 0² - 4 * 1 * 9 = 0 - 36 = - 36
Δ < 0 - brak rozwiązań
4 - 3x = 0
-3x = - 4/:(-3)
x₂ = [tex]\frac{4}{3}[/tex] = [tex]1\frac{1}{3}[/tex]
7 - x²
Δ = 0² - 4* (-1) * 7 = 0 + 28 = 28
√Δ = [tex]\sqrt{28}[/tex] = 2[tex]\sqrt{7}[/tex]
x₃ = [tex]\frac{0 -2\sqrt{7} }{2 * (-1)} = \frac{-2\sqrt{7} }{-2} =\sqrt{7}[/tex]
x₄ = [tex]\frac{0 +2\sqrt{7} }{2 * (-1)} = \frac{2\sqrt{7} }{-2} =-\sqrt{7}[/tex]
Zad. 5
a₆ = a₁ + 5r
a₁₂ = a₁ + 11 r
a₁ + 5r = 10
a₁ + 11r = 12/* (-1)
a₁ + 5r = 10
- a₁ - 11r = - 12
- 6r = - 2/ : (-6)
r = [tex]\frac{2}{6} = \frac{1}{3}[/tex]
a₁ + 5 * [tex]\frac{1}{3}[/tex] = 10
a₁ = 10 - [tex]\frac{5}{3}[/tex] = [tex]\frac{30}{3} - \frac{5}{3} = \frac{25}{3} = 8\frac{1}{3}[/tex]
[tex]a_{n}[/tex] = a₁ + (n - 1) * r = [tex]8\frac{1}{3} + \frac{1}{3}n - \frac{1}{3} = \frac{1}{3} n + 8[/tex]
