Question

1.rozwiąż rownanie x3-3x2-4x+12=0

2. rozwiaz rownanie 3x+4/x+2 = x+8/x+5 (zapis w postaci ułamków)

Answer 1

1.rozwiąż rownanie

x3-3x2-4x+12=0

x²(x - 3) - 4(x - 3) = 0

(x - 3)(x² - 4) = 0

(x - 3)(x - 2)(x  +2) = 0

x - 3 = 0   lub   x - 2 = 0   lub   x + 2 = 0

x = 3         lub       x = 2     lub   x = -2

 

2. rozwiaz rownanie 3x+4/x+2 = x+8/x+5 (zapis w postaci ułamków)

(3x+4) / (x+2) =( x+8) / (x+5)      x ≠ -2   i x ≠ - 5

(x + 2) * (x + 8) = (3x + 4) * (x + 5)

x² + 8x + 2x + 16 = 3x² + 15x + 4x + 20

x² - 3x² + 10x - 19x + 16 - 20 = 0

-2x² - 9x - 4 = 0     /*(-1)

2x² + 9x + 4 = 0

Δ = 81 - 32 = 49

√Δ = 7

x1 = (-9 - 7)/4 = -16/4 = -4

x2 = (-9 + 7)/4 = -2/4 = -1/2

Answer 2

1.

x^3 - 3x^2 -4x+12 = 0

x^2*(x-3)-4(x-3) = 0

(x-3)(x^2-4) = 0

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

x-3 = 0   => x = 3

lub

x+2 = 0   => x = -2 

lub

x-2 = 0   => x = 2

Odp. x = -2   v   x = 2   v   x = 3

 

2.

(3x+4)/(x+2) = (x+8)/(x+5)  I*(x+2)(x+5)                         Z: x+2 ≠ 0   =>  x ≠ -2

                                                                                           x+5 ≠ 0   => x ≠ -5

                                                                                       D = R \ {-5;_2}

(x+2)(x+8) = (x+5)(3x+4)

x^2 +8x+2x+16 = 3x^2 +4x+15x+20

x^2 - 3x^2 +10x-19x+16-20 = 0

-2x^2-9x-4 = 0  I*(-1)

2x^2 + 9x+4 = 0

Δ = b²-4ac = 81-32 = 49

√Δ = √49 = 7 

x1= (-b-√Δ)/2a = (-9-7)/4 = -4

x2 = (-b+√Δ)/2a = (-9+7)/4 = -2/4 = -1/2

Odp. x = -4   v   x = -1/2