a) 3x^4-6x^3-12x^2=0 3x^2(x^2-2x-4)=0 3x^2=0 Δ=20 x=0 x1=1-[latex]sqrt{5}[/latex] x2=1+[latex]sqrt{5}[/latex] b) 10x^5+15x^3=5x^4 10x^5-5x^4+15x^3=0 5x^3(2x^2-x+3)=0 5x^3=0 Δ<0 c) x^3+2x^2=5x+10 x^3+2x^2-5x-10=0 x^2(x+2)-5(x+2)=0 (x^2-5)(x+2)=0 (x-[latex]sqrt{5}[/latex])(x+[latex]sqrt{5}[/latex])(x+2)=0 x=[latex]sqrt{5}[/latex] x=-[latex]sqrt{5}[/latex] x=-2
a) 3x⁴-6x³-12x² = 0 3x²(x²-2x-4) = 0 3x² = 0 => x = 0 lub x²-2x-4 = 0 Δ = b²-4ac = 4+16 = 20 √Δ = √20 = 2√5 x1 = (-b-√Δ)/2a = (2-2√5)/2 = 1-√5 x2 = (-b+√Δ)/2a = (2+2√5)/2 = 1+√5 Odp. x = 1-√5 v x = 0 x = 1+√5 b) 10x⁵+15x³ = 5x⁴ 10x⁵-5x⁴+15x³ = 0 /:5 2x⁵-x⁴+3x³ = 0 x³(2x²-x+3) = 0 x³ = 0 => x = 0 lub 2x²-x+3 = 0 Δ = 1-24 = -23 Δ < 0, brak pierwiastków równania Odp. x = 0 c) x³+2x² = 5x+10 x³+2x²-5x-10 = 0 x²(x+2)-5(x+2) = 0 (x+2)(x²-5) = 0 (x+2)(x+√5)(x-√5) = 0 x+2 = 0 => x = -2 lub x+√5 = 0 => x = -√5 lub x-√5 = 0 => x = √5 Odp. x = -√5 v x = -2 v x = -√5
