Question

Równania kwadratowe
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a) x²-8x+16=0
b) x²-x+1/4=0
c) x²+2x-8=0
d) 3x²-4x+1=0
e) x²-4x+25=0
f) 2x²-3x+12,25=0

Answer 1

A) x²-8x+16=0
Δ=(-8)²-4*1*16
Δ=64-64
Δ=0
x0=8/2*1=4
y=(x-4)²

b) x²-x+1/4=0
Δ=(-1)²-4*1*1/4
Δ=1-1
Δ=0
x0=1/2=½
y=(x-½)²

c) x²+2x-8=0
Δ=2²-4*1*(-8)
Δ=4+32
Δ=36
x1=(-2-6)/2=-4 x2=(-2+6)/2=2
y=(x+4)(x-2)

d) 3x²-4x+1=0
Δ=(-4)²-4*3*1
Δ=16-12
Δ=4
x1=(4-2)/6=⅓ x2=(4+2)/6=1
y=3(x-⅓)(x-1)

e) x²-4x+25=0
Δ=(-4)²-4*1*25
Δ=16-100
Δ=-84 brak rozwiazania

f) 2x²-3x+12,25=0
Δ=(-3)²-4*2*12,25
Δ=9-98
Δ=-89 brak rozwiazania

Answer 2

A) x²-8x+16=0
Δ =b²-4ac
Δ= (-8)² -4*1*16
Δ = 64-64
Δ=0

x₀ = -b/2a
x₀ = -(-8) / 2*1
x₀ = 8/2
x₀ = 4

b) x²-x+1/4=0
Δ =b²-4ac
Δ= (-1)² -4*1*1/4
Δ =1-1
Δ=0

x₀ = -b/2a
x₀ = -(-1) / 2*1
x₀ = 1/2

c) x²+2x-8=0
Δ =b²-4ac
Δ= (2)² -4*1*(-8)
Δ = 4+32
Δ=36
√Δ=√36
√Δ= 6

x₁ = (-b+√Δ) / 2a
x₁ = (-2+6) / 2*1
x₁ = -12/2
x₁ = -6

x₂ = (-b-√Δ) / 2a
x₂ = (-2-6) / 2*1
x₂ = (-8) / 2
x₂ = -4

d) 3x²-4x+1=0
Δ =b²-4ac
Δ= (-4)² -4*3*1
Δ = 16-12
Δ=4
√Δ=√4
√Δ= 2

x₁ = (-b+√Δ) / 2a
x₁ = (4+2) / 2*3
x₁ = 8/6
x₁ = 1 ⅓

x₂ = (-b-√Δ) / 2a
x₂ = (4-2) / 2*3
x₂ = 2 / 6
x₂ = ⅓

e) x²-4x+25=0

Δ =b²-4ac
Δ= (-4)² -4*1*25
Δ = 16-100
Δ=-84
brak rozwiązania

f) 2x²-3x+12,25=0
Δ =b²-4ac
Δ= (-3)² -4*2*12,25
Δ = 9-98
Δ=-89
brak rozwiązania