Question

Rozwiąż równania i nierówności.

a)6x^2-x-2=0

b)(x+2)^2=2x=7

c)x^4-6x^2+5=0

d)x^4-6X^2+8=0

e)9-25x^2>0

f)-x^2-3X<0

g)(A-x)(5-X)<0

Answer 1

a)

 

6x² - x - 2 = 0

 

Δ = 1 - 4*6*( - 2)

Δ = 1 + 48 = 49

√Δ = √49 = 7

 

x₁ = (1 - 7)/12  ∨  x₂ = (1 + 7)/12

 

x₁ = - 6/12  ∨  x₂ = 8/12

 

x₁ = - 1/2  ∨  x₂ = 2/3

 

b)

 

(x + 2)² + 2x = 7

 

x² + 4x + 4 + 2x - 7 = 0

 

x² + 6x + - 3 = 0

 

Δ = 36 - 4*1*( - 3)

Δ = 36 + 12 = 48

√Δ = √48 = √(16*3) = 4√3

 

x₁ = (- 1 - 4√3)/2  ∨  x₂ = (1 + 4√3)/2

 

c)

 

x⁴ - 6x² + 5 = 0

 

(x²)² - 6x² + 5 = 0

 

Niech: x² = t

 

Wtedy:

 

t² - 6t + 5 = 0

 

t² - t - 5t + 5 = 0

 

t(t - 1) - 5(t - 1) = 0

 

(t - 1)(t - 5) = 0

 

t - 1 = 0   ∨  t - 5 = 0

 

t = 1  ∨ t = 5

 

Ale: t = x²,

 

więc:

 

x² = 1   ∨  x²  = 5

 

x² - 1 = 0  ∨  x² - 5 = 0

 

(x - 1)(x + 1) = 0   ∨ (x - √5)(x + √5) = 0

 

x = - 1    ∨ x = 1   ∨ x = - √5   ∨ x = √5

 

d)

 

x⁴ - 6x² + 8 = 0

 

(x²)² - 6x² + 8 = 0

 

Niech: x² = t

 

Wtedy:

 

t² - 6t + 8 = 0

 

Δ = 36 - 32 = 4

√Δ = 2

 

 

x₁ = (6 - 2)/2  ∨  x₂ = (6 + 2)/2

 

x₁ = 4/2  ∨  x₂ = 8/2

 

x₁ = 2  ∨  x₂ = 4

 

e)

 

9 - 25x² > 0

 

- 25(x² - 9/25) > 0 / :( - 25)

 

(x² - 9/25) < 0

 

(x - 0,6)(x + 0,6) < 0

 

x ∈ ( - 0,6 ; 0,6)

 

f)

 

- x² - 3x < 0

 

- x(x + 3) < 0 / * ( - 1)

 

x(x + 3) > 0

 

x∈ ( - ∞ , - 3) U ( 0 , + ∞)

 

g)

 

(A - x)(5 - X) < 0

 

[ - (x - A)]*[ - ( x - 5)] < 0

 

(x - A)(x - 5) < 0

 

1)

 

A > 5

 

x ∈ ( - ∞ , 5)  U ( A , + ∞)

 

2) A < 5

 

x ∈ ( - ∞ , A)  U ( 5 , + ∞)