Question

RÓWNANIA KWADRATOWE;

a. 2x²+7x=0

b. x²-4=0

c.9x²-1=0

d.2x²+x-1=0

e.4x²-1=(2x-1) (x+3)

f.(x-3) (2x-5)=4x²-2x-20

g.2x²+9x-5=(2x-1)²

h.(x-3)(x+3)=7

ROWNANIA I NIEROWNOSCI Z WART. BEZWG.

od 4x² do 9 pod pierwiastkiem rownanie

a.4x²-12x+9 >7

Answer 1

a)

2x² + 7x = 0

x(2x + 7) = 0

x = 0  lub 2x + 7 = 0

x = 0 lub  x = -3,5

----------------------

b)

x² - 4 = 0

(x -2)(x +2) = 0

x = 2 lub x = -2

--------------------------

c)

9x² - 1 = 0

(3x -1)(3x +1) = 0

3x -1 = 0  lub 3x +1 = 0

3x = 1 lub 3x = -1

x = 1/3  lub x = -1/3

=========================

d)

2x² + x - 1= 0

Δ = 1 - 4*2*(-1) = 1 + 8 = 9

√Δ = 3

x1 = [-1 -3]/2 = -2

x2 = [-1 +3]/2 = 1

-----------------------------------

e)

4x² -1 = (2x -1)(x + 3)

4x² - 1 = 2x² +6x -x -3

2x² - 5x  +2 = 0

Δ = 25 -4*2*2 = 25 - 16 = 9

√Δ = 3

x1 = [5 -3]/4 = 1/2

x2 = [5 +3]/4 = 2

-----------------------------------

f)

(x-3)(2x - 5) = 4x²-2x - 20

2x² -5x - 6x + 15 = 4x² - 2x - 20

2x² +9x -35 = 0

Δ = 81 - 4*2*(-35) = 81 + 280 = 361

√Δ = 19

x1 = [ -9 - 19]/4 = - 7

x2 = [-9 +19]/4 = 2,5

-----------------------------------------

g)

2x² + 9x - 5 = (2x -1)²

2x² + 9x - 5 = 4x² - 4x + 1

2x² - 13x +6 = 0

Δ = 169 - 4*2*6 = 169 - 48 = 121

√Δ = 11

x1 = [13 - 11]/4 = 1/2

x2 = [13 +11]/4 = 6

-------------------------------------------

h)

(x-3)(x +3) = 7

x² - 9 = 7

x² - 16 = 0

(x -4)(x +4) = 0

x = 4  lub   x = - 4

--------------------------------------

=======================================

√(4x² -12 x + 9) > 7

podnoszę do kwadratu

4x² - 12 x + 9 > 49

4x² - 12 x - 40 > 0 / : 4

x² - 3x - 10 > 0

Δ = 9 - 4*1*(-10) = 9 + 40 = 49

√Δ = 7

x1 = [3 - 7]/2 = -2

x2 = [3 + 7]/2 = 5

x ∈ ( - ∞ ; -2) u ( 5; + ∞ )

=========================

 

 

 

Answer 2

a) 2x²+7x=0

x(2x+7)=0

x=0 ∨ 2x=-7

x=0 ∨ x=-3,5

 

b) x²-4=0

(x+2)(x-2)=0

x=-2 ∨ x=2

 

c) 9x²-1=0

(3x-1)(3x+1)=0

3x=1 ∨ 3x=-1

x=⅓ ∨ x=-⅓

 

d) 2x²+x-1=0

Δ = 1 + 8 = 9

x₁ = (-1 +3)/2 = 1

x₂ = (-1 -3)/2 = -2

 

e) 4x²-1=(2x-1)(x+3)

(2x-1)(2x+1)=(2x-1)(x+3)

(2x-1)(2x+1)-(2x-1)(x+3)=0

(2x-1)[(2x+1)-(x+3)]=0

(2x-1)(2x+1-x-3)=0

(2x-1)*(x-2)=0

2x-1=0 ∨ x-2=0

x=½ ∨ x=2

 

f)

(x-3)(2x - 5) = 4x²-2x - 20

2x²-5x-6x+15 = 4x²-2x-20

2x² +9x -35 = 0

Δ= 81 + 280 = 361

x₁ =(-9 +19)/4 =2,5

x₂ = (-9-19]/4 = - 7

 

g) 2x²+9x-5=(2x-1)²

2x²+9x-5 = 4x²-4x+ 1

2x²-13x+6 = 0

Δ = 169 - 48 = 121

x₁ =(13 +11)/4 = 6

x₂ =(13 - 11)/4 = ½

 

h) (x-3)(x+3)=7

x²-9-7=0

x²-16=0

(x-4)(x+4)=0

x=4∨ x=-4

 

ROWNANIA I NIEROWNOSCI Z WART. BEZWG.

 

a) √(4x²-12x+9) >7

|2x-3|>7

2x-3>7 ∨ 2x-3<-7

x>5 ∨ x<-2

x∈(-∞;-2) U ((5,+∞)

Answer 3

tylko nie wiem jak zrobic to:

ROWNANIA I NIEROWNOSCI Z WART. BEZWG.

od 4x² do 9 pod pierwiastkiem rownanie

a.4x²-12x+9 >7