Question

rozwiąż równanie

a)(x+1)/2=8/(x-5)

b)(2x+10)/(x+5)=x-2

Answer 1

a) zał. x ≠ 5

(x+1)/2=8/(x-5) 

(x - 5)*(x  +1) = 2 * 8

x² + x - 5x - 5 = 16

x² - 4x - 5 - 16 = 0

x² - 4x - 21 = 0

a = 1, b = -4, c = -21

Δ = b² - 4ac = 16 + 84 = 100

√Δ = 10

x1 = (-b - √Δ) / 2a = (4 - 10) / 2 = -6/2 = -3 ∈ D

x2 = (-b + √Δ) / 2a = (4 + 10) / 2 = 14/2 = 7  ∈ D

odp. x = -3 , x = 7

b)  D:  x ≠ -5 

(2x+10)/(x+5) =  (x-2) / 1

2x + 10) * 1 = (x  +5)(x - 2)

2x + 10 = x² - 2x + 5x - 10

x² + 3x - 10 = 2x + 10

x² + 3x - 2x - 10 - 10 = 0

x ² + x - 20 = 0

Δ = 1 + 80 = 81

√Δ = 9

x1 = (-1 - 9) / 2 = -10 / 2 = -5 ∉ D

x2 = (-1  +9) / 2 = 8 / 2 = 4  ∈ D

odp. x = 4 

 

Answer 2

a)

x≠5

 

(x+1)/2=8/(x-5)     /*2

x+1=16/(x-5)       /*(x-5)

(x+1)(x-5)=16

x^2-5x+x-5=16     /-16

x^2-4x-21=0

Δ=16-4*(-21)*1

Δ=100

√100=10

 

x1=4-10/2=-3

x2=4+10/2=7

 

b)

x≠-5

 

(2x+10)/(x+5)=x-2    /*(x+5)

 

2x+10=(x-2)(x+5)

2x+10=x^2+5x-2x-10

2x+10=x^2+3x-10     /-10

2x=x^2+3x-20         /-2x

x^2+x-20=0

Δ=1-4^(-20)*1

Δ=1+80

Δ=81

√81=9

 

x1=-1-9/2=-5     minus pięć nie może być, ponieważ mamy założenie więc:

x2=-1+9/2=4

 

x=4 ---- porawna odpowiedź.