ZAD 1 Rozłóż wielomiany na czynniki stopnia co najwyżej drugiego: A) W(x)+x³-7x²+6x=x(x²-7x+6)=x(x-6)(x-1) Δ=49-24=25 √Δ=5 x₁=7+5/2=12/2=6 x₂=7-5/2=2/2=1 (x-6)(x-1) B) W(x)=x⁶+7x⁵+6x⁴=x⁴(x²+7x+6)=x⁴(x+6)(x+1) Δ=49-24=25 √Δ=5 x₁=-7+5/2=-2/2=-1 x₂=-7-5/2=-12/2=-6 (x+6)(x+1) ZAD 2 a) log (3x-9)- log (30-x)= 0 D: 3x-9>0 3x>9 x>3 30-x>0 x<30 D: x∈(3; 30) log(3x-9)/(30-x)=log1 (3x-9)/(30-x)=1 3x-9=30-x 4x=39 x=39/4=9,75 ∈D b) log₂ x+log₂(x+2) =log₂(x+6) log₂x(x+2)=log₂(x+6) D: x+2>0 x>-2 x+6>0 x>-6 D x∈(-2; ∞) x(x+2)=x+6 x²+2x=x+6 x²+x-6=0 Δ=1+24=25 √Δ=5 x₁=-1+5/2=4/2=2 ∈D x₂=-1-5/2=-6/2=-3 ∉D
zad. 1 (chyba zabrakło znaku "=") A) W(x) = x³-7x²+6x = x(x² - 7x + 6) = x(x - 1)(x - 6) Δ = (-7)² - 4*6 = 49 - 24 = 25 x₁ = (7 - 5) : 2 = 1 lub x₂ = (7 + 5) : 2 = 6 B) W(x) = x⁶+7x⁵+6x⁴ = x⁴(x² - 7x + 6) = x²(x - 1)(x - 6)x² zad. 2 a) log (3x-9) - log (30-x)= 0 (!) 3x - 9 >0 i 30 - x >0 3 < x < 30 log (3x-9) = log (30-x) 3x - 9 = 30 - x 4x = 39 x = 9¾ b) log₂x+log₂(x+2) =log₂(x+6) (!) x > 0 log₂[x(x+2)] =log₂(x+6) x(x + 2) = x + 6 x² + 2x = x + 6 x² + x - 6 = 0 Δ = 1 + 4 * 6 = 25 (x₁ = -3 lub x₂ = 2 ) i x > 0 x = 2
