1) [latex]4x^3 - x^2 - 16x + 4 = 0[/latex] [latex]x^2 cdot (4x - 1) - 4 cdot (4x - 1) = 0[/latex] [latex](4x - 1)(x^2- 4) = 0[/latex] [latex](4x - 1)(x- 2)(x + 2) = 0[/latex] [latex]4x - 1 = 0 vee x- 2 = 0 vee x + 2 = 0[/latex] [latex]4x - 1 = 0 [/latex] [latex]4x = 1 / : 4[/latex] [latex]x = frac{1}{4}[/latex] [latex]x- 2 = 0 [/latex] [latex]x = 2[/latex] [latex]x + 2 = 0 [/latex] [latex]x = - 2[/latex] Odp. [latex]x = - 2 lub x = frac{1}{4} lub x = 2[/latex] 2) [latex]x^5 - 5x^3 + 4x = 0[/latex] [latex]x cdot (x^4 - 5x^2 + 4) = 0[/latex] [latex]x cdot (x^2 - 1)(x^2 - 4) = 0[/latex] [latex]x cdot (x - 1)(x + 1)(x - 2)(x+2) = 0[/latex] [latex]x = 0 vee x - 1 = 0 vee x + 1 = 0 vee x - 2 = 0 vee x+2 = 0[/latex] [latex]x = 0[/latex] [latex]x - 1 = 0[/latex] [latex]x = 1[/latex] [latex]x + 1 = 0[/latex] [latex]x = - 1[/latex] [latex]x - 2 = 0 [/latex] [latex]x = 2[/latex] [latex]x+2 = 0[/latex] [latex]x = -2[/latex] Odp. [latex]x = - 2 lub x = - 1 lub x = 0 lub x = 1 lub x = 2[/latex] ----------------------------------------------------------------------------------------------- [latex]x^4 - 5x^2 + 4 =_{x^2 = t; t geq 0} = t^2 - 5t + 4 = (t -1)(t-4) =_{t =x^2} = [/latex] [latex]= (x^2-1)(x^2-4)[/latex] [latex]t^2 - 5t + 4[/latex] [latex]Delta = (-5)^2 - 4 cdot 1 cdot 4 = 25 - 16 = 9; sqrt{Delta} = 3[/latex] [latex]t_1 = frac{5 - 3}{2 cdot 1} = frac{2}{2} = 1[/latex] [latex]t_2 = frac{5 + 3}{2 cdot 1} = frac{8}{2} = 4[/latex] [latex]t^2 - 5t + 4 = 1 cdot (t - 1)(t - 4) = (t - 1)(t - 4)[/latex] ----------------------------------------------------------------------------------------------- 3) [latex]8x^3 - 12x^2 - 20x + 30 =0[/latex] [latex]4x^2 cdot (2x - 3) - 10 cdot (2x - 3) =0 [/latex] [latex](2x - 3)(4x^2 - 10)=0 [/latex] [latex](2x - 3)[(2x)^2 - (sqrt{10})^2]=0 [/latex] [latex](2x - 3)(2x - sqrt{10})(2x + sqrt{10})=0 [/latex] [latex]2x - 3 = 0 vee 2x - sqrt{10} = 0 vee 2x + sqrt{10}=0[/latex] [latex]2x - 3 = 0[/latex] [latex]2x = 3 / :2[/latex] [latex]x = frac{3}{2}[/latex] [latex]x = 1frac{1}{2}[/latex] [latex]2x - sqrt{10} = 0[/latex] [latex]2x = sqrt{10} / :2[/latex] [latex]x = frac{sqrt{10}}{2}[/latex] [latex]2x + sqrt{10} = 0 [/latex] [latex]2x = -sqrt{10} / :2[/latex] [latex]x = -frac{sqrt{10}}{2}[/latex] Odp. [latex]x = -frac{sqrt{10}}{2} lub x = 1frac{1}{2} lub x = frac{sqrt{10}}{2}[/latex] 4) [latex]frac{1}{16}x^7 - 25x^3 = 0[/latex] [latex]x^3 cdot (frac{1}{16}x^4 - 25) = 0[/latex] [latex]x^3 cdot [(frac{1}{4}x^2)^2 - 5^2] = 0[/latex] [latex]x^3 cdot (frac{1}{4}x^2 - 5)(frac{1}{4}x^2 + 5) = 0 [/latex] [latex]x^3 cdot [(frac{1}{2}x)^2 - (sqrt{5})^2](frac{1}{4}x^2 + 5) = 0 [/latex] [latex]x^3 cdot (frac{1}{2}x- sqrt{5})(frac{1}{2}x+ sqrt{5})(frac{1}{4}x^2 + 5) = 0 [/latex] [latex]x^3= 0 vee frac{1}{2}x- sqrt{5} = 0 vee frac{1}{2}x+ sqrt{5} = 0 vee frac{1}{4}x^2 + 5 = 0 [/latex] [latex]x^3= 0[/latex] [latex]x = 0[/latex] [latex]frac{1}{2}x- sqrt{5} = 0[/latex] [latex]frac{1}{2}x = sqrt{5} / cdot 2[/latex] [latex]x = 2sqrt{5}[/latex] [latex]frac{1}{2}x+ sqrt{5} = 0[/latex] [latex]frac{1}{2}x = -sqrt{5} / cdot 2[/latex] [latex]x = -2sqrt{5}[/latex] [latex]frac{1}{4}x^2 + 5 = 0[/latex] [latex]Delta = 0^2 - 4 cdot frac{1}{4} cdot 5 = 0 - 5 = - 5 < 0[/latex] Równanie nie ma rozwiązań: [latex]x in emptyset[/latex] Odp. [latex]x = -2sqrt{5} lub x = 0 lub x = 2sqrt{5}[/latex]
