a) [latex]y = -x^2 + 6x - 5[/latex] [latex]Delta = b^2 - 4ac = 36 - 20 = 16[/latex] [latex]x_{1/2} = frac{-b pm sqrt{Delta}}{2a} = frac{-6 pm sqrt{16}}{-2} = frac{6 pm 4}{2} = 3 pm 2[/latex] [latex]x_1 = 3 + 2 = 5[/latex] [latex]x_2 = 3 - 2 = 1[/latex] [latex]W = (p, q)[/latex] [latex]p = frac{-b}{2a} = frac{-6}{-2} = 3[/latex] [latex]q = frac{-Delta}{4a} = frac{-16}{-4} = 4[/latex] Ramiona w dół, wierzchołek w [latex]W = (3, 4)[/latex], miejsca zerowe w 1 i 5. Rosnąca [latex](-infty, 3>[/latex], malejąca w [latex]<3, +infty[/latex]. b) [latex]x^2 - 64 = 0[/latex] [latex]x^2 = 64[/latex] [latex]x = pm 8[/latex] [latex]x^2 - 3x = 0[/latex] [latex]x(x - 3) = 0[/latex] [latex]x = 0 x - 3 = 0 aightarrow x = 3[/latex] c) strzelam w znak < [latex]x^2 < 5x - 4[/latex] [latex]x^2 - 5x + 4 < 0[/latex] [latex]Delta = b^2 - 4ac = 25 - 16 = 9[/latex] [latex]x_{1/2} = frac{-b pm sqrt{Delta}}{2a} = frac{5 pm 3}{2}[/latex] [latex]x_1 = frac{5 + 3}{2} = 4[/latex] [latex]x_2 = frac{5 - 3}{2} = 1[/latex] [latex](x - 1)(x - 4) < 0[/latex] [latex]x - 1 < 0 ightarrow x < 1 x - 4 < 0 ightarrow x < 4[/latex] [latex]x in (1, 4)[/latex] [latex]x^2 - 10x ge -9[/latex] [latex]x^2 - 10x + 9 ge 0[/latex] [latex]Delta = b^2 - 4ac = 100 - 36 = 64[/latex] [latex]x_{1/2} = frac{-b pm sqrt{Delta}}{2a} = frac{10 pm 8}{2} = 5 pm 4[/latex] [latex]x_1 = 5 + 4 = 9[/latex] [latex]x_2 = 5 - 4 = 1[/latex] [latex](x - 1)(x - 9) ge 0[/latex] [latex]x - 1 ge 0 ightarrow x ge 1 x - 9 ge 0 ightarrow x ge 9[/latex] [latex][/latex] Załączniki: 1 - zad a) 2 - zad c_1) 3 - zad c_2)
