2. h) [latex]-0,5x^2-sqrt2x+3<0[/latex] [latex]Delta=(-sqrt2)^2-4cdot(-0,5)cdot3=2+6=8[/latex] [latex]sqrt{Delta}=sqrt8=2sqrt2[/latex] [latex]x_1=frac{sqrt2-2sqrt2}{-1}=sqrt2[/latex] [latex]x_2=frac{sqrt2+2sqrt2}{-1}=-3sqrt2[/latex] Parabola z ramionami skierowanymi do dołu [latex]xin(-infty;-3sqrt2)cup(sqrt2;+infty)[/latex] ================= a) [latex](6-x)(x-4)<(x-1)(x-4)[/latex] [latex](6-x)(x-4)-(x-1)(x-4)<0[/latex] [latex](x-4)(6-x-x+1)<0[/latex] [latex](x-4)(-2x+7)<0[/latex] [latex]-2(x-4)(x-3,5)<0[/latex] Parabola z ramionami skierowanymi do dołu [latex]xin(-infty;3,5)cup(4;+infty)[/latex] ---------------- b) [latex](x+1)^3ge(x+1)(x^2-x+2)[/latex] [latex](x+1)^3-(x+1)(x^2-x+2)ge0[/latex] [latex]x^3 + 3x^2 + 3x + 1-(x^3-x^2+2x+x^2-x+2)ge0[/latex] [latex]x^3 + 3x^2 + 3x + 1-x^3+x^2-2x-x^2+x-2ge0[/latex] [latex]3x^2 + 2x - 1ge0[/latex] [latex]Delta=2^2-4cdot3cdot(-1)=4+12=16[/latex] [latex]sqrt{Delta}=sqrt{16}=4[/latex] [latex]x_1=frac{-2-4}{6}=frac{-6}{6}=-1[/latex] [latex]x_2=frac{-2+4}{6}=frac{2}{6}= frac{1}{3}[/latex] Parabola z ramionami skierowanymi do góry [latex]xinleft(-infty;-1 ight angle cupleftlangle frac{1}{3} ;+infty ight)[/latex]
