Question

Rozwiąż równania kwadratowe:

1. x^2=10+3x

2. 1-x^2=0

3. -x^2-12x-35=0

4. (3-5x)(2x+1)=(2x+1)(3x+2)

5. (1-x)^2=1

6. x^2=2-x

7. 4-x^2=0

8. 4x^2+4x+1=0

9. (2x-1)^2=16

10. (x-3)(2x+5)=(2x-6)(2x-1)

x^2 to x podniesione do kwadratu, potrzebne koniecznie na dziś.

Answer 1

1. x^2=10+3x

    x^2-3x-10=0

a=1

b=-3

c=-10

Obliczanie delty ze wzoru: b^2-4ac

(-3)^2-4*1* (-10)=9+40=49

pierwiastek z 49=7

I liczysz x1 i x2 ;)

x1=3-7całość dzielone przez 2= -4/2= -2

x2=3+7 całość dzielone przez 2= 10/2=5

 

 

2. 1-x^2=0

a=-1

b=0

c=1

delta: -4*(-1)*1=4

pierwiastek z 4=2

x1=-2/-2= 1

x2=2/-2= -1

 

3. -x^2-12x-35=0

a=-1

b=-12

c=-35

delta: 144-4*(-1)*(-35)=144-140=4

pierwiastek z 4=2

x1=12-2 całość przez -2= 10/-2= -5

x2=12+2całośc przez -2= 14/-2=-7

 

4. (3-5x)(2x+1)=(2x+1)(3x+2)

6x+3-10x^2-5x=6x^2+4x+3x+2

6x+3-10x^2-5x-6x^2-4x-3x-2=0

-16x^2-6x+1=0

a=-16

b=-6

c=1

delta: 36-4*(-16)*1=36+64=100

pierwiastek z 100=10

x1=6-10 całość przez  -32= -4/-32=1/8

x2=6+10 całość przez -32= 16/-32=- 1/2

 

5.(1-x)(1-x)=1

1-x-x+x^2-1=0

x^2-2x=0

a=1

b=-2

c=0

delta: 4-0=4

pierwiastek z 4=2

x1= 2-2całość przez 2=0

x2=2+2 całość przez 2= 4/2=2

 

Z kolejnymi nie powinno być już problemu ;)

wszystko analogicznie do tych przykładów ;)
Pozdrawiam ;) 

 

Answer 2

zad.1

x² = 10 + 3x

x² - 3x – 10 = 0

∆ = b² - 4ac

∆ = (-3)² - 4 * 1 * (-10) = 9 + 40 = 49

√∆ = √49 = 7

x₁ = (-b-√∆)/2a = (3 -7)/2 = -4/2 = -2

x₂ = (-b+√∆)/2a = (3 +7)/2 = 10/2 = 5

 

zad.2

1 - x² = 0

-x² +1 = 0  /* (-1)

x² - 1 = 0

∆ = b² - 4ac

∆ = - 4 * 1 * (-1) = 4

√∆ = √4 = 2

x₁ = (-b-√∆)/2a =   -2/2 = -1

x₂ = (-b+√∆)/2a =  2/2 =  1

 

zad.3

-x² -12x – 35 = 0  /* (-1)

x² + 12x + 35 = 0

∆ = b² - 4ac

∆ = 12² - 4 * 1 * 35 = 144 + 140 = 4

√∆ = √4 = 2

x₁ = (-b-√∆)/2a = (-12 -2)/2 = -14/2 = -7

x₂ = (-b+√∆)/2a = (-12 +2)/2 = -10/2 =- 5

 

zad.4

(3-5x)(2x+1) = (2x+1)(3x+2)

6x + 3 -10x² – 5x = 6x² +4x +3x + 2

6x + 3 -10x² – 5x - 6x² -4x -3x – 2 = 0

-16x² -6x + 1 = 0   /* (-1)

16x² + 6x – 1 = 0

∆ = b² - 4ac

∆ = 6² - 4 * 16 * (-1) = 36 + 64= 100

√∆ = √100 = 10

x₁ = (-b-√∆)/2a = (-6 -10)/32 = -16/32 = - ½

x₂ = (-b+√∆)/2a = (-6 +10)/32 = 4/32 = 1/8

 

zad.5

(1 –x)² = 1

1 – 2x + x² -1 = 0

x² - 2x = 0

∆ = b² - 4ac

∆ = (-2)² - 4 * 1 * 0 =  4

√∆ = √4 = 2

x₁ = (-b-√∆)/2a = (2 -2)/2 = 0

x₂ = (-b+√∆)/2a = (2 +2)/2 = 2

 

zad.6

x² = 2 – x

x² + x – 2 = 0

∆ = b² - 4ac

∆ =  1² - 4 * 1 * (-2) =  1+8 = 9

√∆ = √9 = 3

x₁ = (-b-√∆)/2a = (-1 -3)/2 = -4/2 = -2

x₂ = (-b+√∆)/2a = (-1 +3)/2 = 2/2 = 1

 

zad. 7

4 - x² = 0

-x² + 4 = 0  /* (-1)

x² - 4 = 0

∆ = b² - 4ac

∆ = - 4 * 1 * (-4) =  16

√∆ = √16 = 4

x₁ = (-b-√∆)/2a =   -4/2 =  -2

x₂ = (-b+√∆)/2a = 4/2 = 2

 

zad.8

4x² + 4x + 1 = 0

∆ = b² - 4ac

∆ = 4² - 4 * 4 * 1 =  16-16 = 0

√∆ =0 jest jeden pierwiastek

x₁ = x₂ = -b/2a = -4/8 = - ½

 

zad.9

(2x-1)² = 16

4x² - 4x + 1 – 16 = 0

4x² -4x -15 = 0

∆ = b² - 4ac

∆ =  (-4)² - 4 * 4* (-15) =  16+  240=  256

√∆ = √256 = 16

x₁ = (-b-√∆)/2a = (4 -16)/2 = -14/2 = -7

x₂ = (-b+√∆)/2a = (4 +16)/2 = 20/2 = 10

 

zad.10

(x-3)(2x+5) = (2x-6)(2x-1)

2x² + 5x – 6x – 15 = 4x² - 2x – 12x + 6

2x² + 5x – 6x – 15 - 4x² + 2x + 12x -  6 = 0

-2x² + 13x -21 = 0  /* (-1)

2x² - 13x + 21 = 0

∆ = b² - 4ac

∆ =  13² - 4 * 2*21 =  169 – 168 = 1

√∆ = √1 = 1

x₁ = (-b-√∆)/2a = (13 -1)/4=  12/4 = 3

x₂ = (-b+√∆)/2a = (13 +1)/4=14/4  = 7/2 = 3 ½