1.
a) x²- 6x + 10 = 0
Δ = 36 -4*10 = 36 - 40 = -4 < 0
równanie sprzeczne
b) 4x² - 4x + 1 = 0
Δ = 16 - 4*4 = 16-16 = 0
x = 1/2
c) x² + 6x + 7 = 0
Δ = 36 - 4*7 = 36 - 28 = 8
√Δ = √8 = 2√2
x₁ = (-6 + 2√2)/2 = √2 - 3
x₂ = (-6 - 2√2)/2 = -√2 - 3
2.
f(x) = 2x² - x + 1
f(2-x) = g(x) = 2(2-x)² - (2-x) + 1 = 2(4 - 4x + x²) - 2 + x + 1 = 8 - 8x + 2x² - 1 + x = 2x² - 7x + 7
f(2x) = h(x) = 2(2x)² - 2x + 1 = 2 * 4x² - 2x + 1 = 8x² - 2x + 1
f(2-x) = f(2x)
2x² - 7x + 7 = 8x² - 2x + 1
-6x² - 5x + 6 = 0
Δ = 25 - 4 * 6 * -6 = 25 + 144 = 169
√Δ =13
x₁ = (5-13)/-12 = -8/-12 = -2/3
x₂ = (5+13)/-12 = 18/-12 = -3/2
3.
a) x² + 2x + 8 ≤ 0
Δ = 4 - 4*8 = 4-32 = -28 < 0 ; brak miejsc zerowych
a > 0
q = 28/4 = 7 > 0
x ∈ ∅
b)
-x² + 4x + 32 < 0
Δ = 16 - 4*32*-1 = 16 + 128 = 144
√Δ = 12
x₁ = (-4+12) / -2 = 8/-2 = -4
x₂ = (-4- 12) / -2 = -16/-2 = 8
x ∈ (-∞,-4)∪(8,+∞)
c) 4x² + 4x + 1 > 0
Δ = 16 - 4*4 = 16-16 = 0
x = -4/8 = -1/2
x ∈ (-∞,-1/2)∪(1/2,+∞)
