Question

1.Rozwiąż równanie.

a)x^ -4x-5=0

b)3x^ -14x-5=0

c)4x^ -2x+¼=0

d)5x^ +2x+1=0

e)7x^ +2x=1

f)-6x^ +x+1=0

2.Rozwiąż rónanie.

a)x(x-5)=2x(x-1)

b)2(1-5x)=(1-x)^

c)x^+4/5=1-x/2

d)(x+1)^/3=(x+4)^/7-1

^-do potęgi drugiej

Answer 1

Rozwiąż równanie.

a)x²-4x-5=0

Δ = b² - 4ac = (-4)² - 4 *1*(-5) = 16 + 20 = 36

√Δ = √36 = 6

x₁ = (-b-√Δ)/2a = ( 4 -6)/2 = -2/2 = -1

x₂ = (-b+√Δ)/2a = ( 4 +6)/2 = 10/2 = 5

 

b)3x²-14x-5=0

 Δ = b² - 4ac = (-14)² - 4 *3*(-5) = 196 + 60 = 256

√Δ = √256 =16

x₁ = (-b-√Δ)/2a = ( 14 -16)/6 = -2/6 = -1/3

x₂ = (-b+√Δ)/2a = ( 14 +16)/2 = 30/2 = 15

 

c)4x²-2x+¼=0    /*4

16x² - 8x + 1 = 0

 Δ = b² - 4ac = (-8)² - 4 *16*1 = 64 -64 = 0

x₁ = x₂ = -b/2a = 8/2 = 4

 

d)5x²+2x+1=0

 Δ = b² - 4ac = 2² - 4 *5*1 = 4 – 20 = -16

Brak rozwiązania bo delta ujemna

 

e)7x²+2x=1

7x² + 2x – 1 = 0

 Δ = b² - 4ac =2² - 4 *7*(-1) = 4 + 28 = 32

√Δ = √32 =√1 6*2 = 4√2

x₁ = (-b-√Δ)/2a = ( -2 -4√2)/14 = -2(1+2√2)/14 = - (1+ 2√2)/7

x₂ = (-b+√Δ)/2a = ( -2 +4√2)/14 = 2(2√2 -1)/14 =  ( 2√2 + 1)/7

 

f)-6x²+x+1=0

Δ = b² - 4ac = 1² - 4 *(-6)*1 = 1 + 24 = 25

√Δ = √25 = 5

x₁ = (-b-√Δ)/2a = ( -1 -5)/-12  = -6/12 =  1/2  

x₂ = (-b+√Δ)/2a = ( -1 +5)/-12 = - 4/12 = - 1/3

 

 

2.Rozwiąż rónanie.

  a)x(x-5)=2x(x-1)

x² - 5x = 2x² - 2x

x² - 5x -2x² + 2x = 0

-x² - 3x = 0

-x (x – 3) = 0

x = 0  lub  x-3 = 0

x = 0 lub  x = 3

 

b)2(1-5x)=(1-x)²

2 – 10x = 1 – 2x - x²

2 – 10x -1 + 2x + x² = 0

x² - 8x + 1 = 0

Δ = b² - 4ac = (-8)² - 4 *1* = 64 -4 =60

√Δ = √60 = √4*15 = 2√15

x₁ = (-b-√Δ)/2a = ( 8 -2√15)/2 = 2(4 -√15)/2 = 4 -√15

x₂ = (-b+√Δ)/2a = ( 8 +2√15)/2 = 2(4 +√15)/2 = 4 +√15

 

 c)x²+4/5=1-x/2 

2(x²+4) = 5(1-x)

2x² + 8 = 5 – 5x

2x² + 5x + 3 = 0

Δ = b² - 4ac = 5² - 4 *2*3= 25- 24 = 1

√Δ = √1 = 1

x₁ = (-b-√Δ)/2a = ( -5 -1)/4  = -6/4 = - 1 1/2

x₂ = (-b+√Δ)/2a = ( -5 +1)/4  = -4/4 = - 1

 

d)(x+1)²/3=(x+4)²/7- 1    /* 21

7(x+1)²=3(x+4)²  - 21

7( x² + 2x + 1) = 3(x² + 8x + 16) -21

7x² + 14x + 7 = 3x² + 24x + 48 -21

7x² + 14x + 7 - 3x² - 24x- 48 + 21 = 0

4x² - 10x –20 = 0

Δ = b² - 4ac = (-10)² - 4 *4*(-20)= 100 + 320= 420

√Δ = √420 =√4*105 = 2√105

x₁ = (-b-√Δ)/2a = (10 -2√105)/8  = 2(5 -√105)/8  = (5 -√105)/4

x₂ = (-b+√Δ)/2a = (10 +2√105)/8  = 2(5 +√105)/8  = (5 +√105)/4