Question

Rozwiąż:

a) x^2-3 1/2x= -1 1/2
b) 3x^2-27=0
c) 16-(x+5)^2=0
d) 4x^2-4x+1<0
e) 7>x^2+6x
f) (4x-2) (x-2) -5(x-2) x=0

f) można, ale nie trzeba rozwiązywać

Proszę o wytłumaczenie, dziękuję.

Answer 1

A)
x² - 3 1/2x + 1 1/2 = 0 /*2
2x² - 7x + 3 = 0
Δ - (-7)² - 4*2*3 = 49 - 24 = 25
√Δ = 5
x₁ = 7-5/4 = 1/2
x₂ = 7+5/4 = 3

b)
3x² - 27 = 0
3x² = 27 /:3
x² = 9 /√
x₁ = 3 lub x₂ = -3

c)
16 - (x + 5)² = 0
16 - x² - 10x - 25 = 0
-x² - 10x - 9 = 0
Δ = (-10)² - 4*(-1)*(-9) = 100 - 36 = 64
√Δ = 8
x₁ = 10-8/-2 = -1
x₂ = 10+8/-2 = -8

d)
4x² - 4x + 1 < 0
Δ = (-4)² - 4*4*1 = 16 - 16 = 0
x₀ = 4/8 = 1/2

a=4, a > 0. ramiona paraboli skierowane ku górze

x ∈ Ф

e)
x² + 6x - 7 < 0
Δ = 6² - 4*1*(-7) = 36 + 38 = 64
√Δ = 8
x₁ = -6-8/2 = -7
x₂ = -6+8/2 = 1

a = 1, a > 0, ramiona paraboli skierowane ku górze

x ∈ (-7; 1)

f)
(4x - 2)(x - 2) - 5(x - 2)x = 0
4x² - 8x - 2x + 4 - 5x² + 10x = 0
-x² + 4 = 0
4 - x² = 0
(2 - x)(2 + x) = 0
x₁ = 2    lub x₂ = -2

Answer 2

a)
x² - 7/2x + 3/2= 0
2x²-7x+3 = 0
Δ = 49-24=25
√Δ =5
x₁ =1/2
x₂ =3

b)
3x²-27 = 0
x²-9=0
(x-3)(x+3)=0
x₁=3 lub x₂= -3

c)
16-x²-10x-25=0
-x²-0x-9=0
Δ =100-36 = 64
√Δ = 8
x₁= -1
x₂= -8

d)
4x² - 4x + 1 < 0
Δ= 16-16=0
x₀=4/8
z zbior pusty

e)
-x²-6x+7>0
x²+6x-7<0
Δ36+38=64
√Δ = 8
x₁ = -7
x₂ = 1

x ∈ (-7; 1)

f)
4x²-8x-2x+4-5x²+10x=0
-x²+4=0
-(x-2)(x+2)
x=2 lub x=-2