Question

Rozwiąż równanie 3(x2-4X)-(X2-4X)2+10=0

Zapisz pierwiastki w kolejności rosnącej.

Wskazówka: Wprowadz niewiadomą pomocniczą t=x2-4x

x2-do kwadratu

)2-nawias do kwadratu

Answer 1

3(x²-4x)-(x²-4x)²+10 = 0      t = x²-4x

3t-t²+10 = 0

-t²+3t+10 = 0

Δ = 3²-4·(-1)·10 = 9+40 = 49

√Δ = √49 = 7

t₁ = (-3-7)/2·(-1) = (-10)/(-2) = 5

t₂ = (-3+7)/2·(-1) = 4/(-2) = -2

 

5 = x²-4x                                             ∨   -2 = x²-4x

x²-4x-5 = 0                                          ∨   x²-4x+2 = 0

Δ = (-4)²-4·1·(-5) = 16+20 = 36                   Δ = (-4)²-4·1·2 = 16-8 = 8

√Δ = √36 = 6                                              √Δ = √8 = 2√2

x₁ = (4-6)/2 = -1                  ∨                    x₁ = (4-2√2)/2 = 2-√2

x₂ = (4+6)/2 = 5                   ∨                    x₂ = (4+2√2)/2 = 2+√2

 

x∈{-1; 2-√2; 2+√2; 5}