Question

Proszę o dokładne rozpisanie 1.rozwiąż równanie, ( =<)-większe bądź równe

a) 6x-9=xkw

b) 4x-4xkw=1

c) xkw+pierwiastek2x-4=0

d) pierwiastek3xkw-9x+6pierwiastek3=0

nierówność e) 6xkw+3x>=0

f) 2xkw-2+3x=<0

g) 2xkw+5x-3>0

h) -3xkw-2x+1=<0

Answer 1

a)  

6x-9=xkw

x² - 6x + 9 = 0

(x - 3)² = 0

x - 3 = 0

x = 3

 

b)

4x-4x² = 1

4x² - 4x + 1 = 0

(2x - 1)² = 0

2x - 1 = 0

2x = 1

x = 1/2

 

c)

x² + √2x - 4 = 0

Δ = 2 - 4 * (-4) = 2 + 16 = 18

√Δ = 3√2

x1 = (-√2 - 3√2)/2 = -4√2/2 = -2√2

x2 = (-√2 + 3√2)/2 = 2√2/2 = √2

 

d)

√3 x² - 9x + 6√3 = 0

Δ = 81 - 4 * 3 * 6 = 81 - 72 = 9

√Δ = 3

x1 = (9 - 3)/√3 = 6/√3 = 6√3/3 = 2√3

x2 = (9 + 3)/√3 = 12/√3 = 12√3/3 = 4√3

 

e)

6x² + 3x ≥ 0

3x(2x + 1) ≥ 0

x(2x  +1) ≥ 0

 

+++++                      +++++

----------o-----------o-------->

           -1/2 ------  0

 

od. x ∈ (-oo, - 1/2) U (0, +oo)

 

f)

2x² - 2 + 3x ≤ 0

2x² + 3x - 2 ≤ 0

Δ = 9 - 4 * 2 * (-2) = 9 + 16 = 25

√Δ = 5

x1 = (-3 - 5)/4 = -8/4 = -2

x2 = (-3 + 5)/4 = 2/4 = 1/2

 

++++++                      ++++++

-----------o------------o---------->

            -2  --------   1/2

x ∈ (-2, 1/2)

 

g)    

2x² + 5x - 3 > 0

Δ = 25 + 24 = 49

√Δ = 7

x1 = (-5 - 7)/4 = -12/4 = -3

x2 = (-5 + 7)/4 = 2/4 = 1/2

 

++++++                      ++++++

-----------o------------o---------->

            -3  --------   1/2

 

x ∈ (-oo, -3) u (1/2, +oo)

 

h)

-3x² - 2x + 1 ≤ 0

3x² + 2x - 1 ≥ 0

Δ = 4 + 12 = 16

√Δ = 4

x1 = (-2 - 4)/6 = -6/6 = -1

x2 = (-2 + 4)/6 = 2/6 =1/3

 

+++++                      ++++++

-----------o------------o---------->

            -1  --------   1/3

x ∈ (-oo, -1) u (1/3, +oo)