a) [latex]frac{3x^2 - 9x}{2} - frac{12}{x^2-3x} = 3[/latex] Zał. [latex]x^2-3x eq 0 \ x cdot (x - 3) eq 0 \ x eq 0 wedge x - 3 eq 0 \\ x eq 0 \\ x - 3 eq 0 \ x eq 3 \\ D = R setminus {0; 3 }[/latex] [latex]frac{3x^2 - 9x}{2} - frac{12}{x^2-3x} = 3 \\ frac{3 cdot (x^2 - 3x)}{2} - frac{12}{x^2-3x} = 3 \\ t = x^2 - 3x wedge x^2 - 3x eq 0 => t eq 0 \\ frac{3t}{2} - frac{12}{t} = 3 \\ frac{3t}{2} - frac{12}{t} - 3 = 0 \\ frac{3t^2}{2t} - frac{24}{2t} - frac{6t}{2t} = 0 \\ frac{3t^2 - 24 - 6t}{2t} = 0 / cdot 2t \\ 3t^2 - 6t - 24 = 0 /:3 \\ t^2 - 2t - 8 = 0[/latex] [latex]Delta = (-2)^2 - 4 cdot 1 cdot (-8) = 4 + 32 = 36; sqrt{Delta} = sqrt{36} = 6 \\ t_1 = frac{2 - 6}{2 cdot 1} = frac{-4}{2} = - 2 \\ t_2 = frac{2 + 6}{2 cdot 1} = frac{8}{2} = 4 \\ t = x^2 - 3x wedge t = -2 \ x^2 - 3x = - 2 \ x^2 - 3x + 2 = 0 \ Delta = (-3)^2 - 4 cdot 1 cdot 2 =9 -8 = 1; sqrt{Delta} = sqrt{1} = 1 \\ x_1 = frac{3 - 1}{2 cdot 1} = frac{2}{2} = 1 in D \\ x_2 = frac{3 + 1}{2 cdot 1} = frac{4}{2} = 2 in D[/latex] [latex]t = x^2 - 3x wedge t = 4 \ x^2 - 3x = 4 \ x^2 - 3x - 4 = 0 \ Delta = (-3)^2 - 4 cdot 1 cdot (-4) =9 +16 = 25; sqrt{Delta} = sqrt{25} = 5 \\ x_1 = frac{3 - 5}{2 cdot 1} = frac{-2}{2} = -1 in D \\ x_2 = frac{3 + 5}{2 cdot 1} = frac{8}{2} = 4 in D[/latex] Stąd: [latex]x in {-1; 1; 2; 4 }[/latex] b) [latex]x^2 - 4x - frac{15}{x^2 - 4x} = 2[/latex] Zał. [latex]x^2 - 4x eq 0 \ x cdot (x - 4) eq 0 \ x eq 0 wedge x - 4 eq 0 \\ x eq 0 \\ x - 4 eq 0 \ x eq 4 \\ D = R setminus {0; 4 }[/latex] [latex]x^2 - 4x - frac{15}{x^2 - 4x} = 2 \\ t = x^2 - 4x wedge x^2 - 4x eq 0 => t eq 0 \\ t - frac{15}{t} = 2 \\ t - frac{15}{t} - 2 = 0 \\ frac{t^2}{t} - frac{15}{t} - frac{2t}{t} = 0 \\ frac{t^2 - 15 - 2t}{t} = 0 / cdot t \\ t^2 - 2t - 15 = 0 \\ Delta = (- 2)^2 - 4 cdot 1 cdot (-15) = 4 + 60 = 64; sqrt{Delta} = sqrt{64} = 8 \\ t_1 = frac{2 - 8}{2 cdot 1} = frac{- 6}{2} = -3 \\ t_2 = frac{2 + 8}{2 cdot 1} = frac{10}{2} = 5[/latex] [latex]t = x^2 - 4x wedge t = -3 \ x^2 - 4x = -3 \ x^2 - 4x +3 = 0 \ Delta = (-4)^2 - 4 cdot 1 cdot 3 = 16 -12 = 4; sqrt{Delta} = sqrt{4} = 2 \\ x_1 = frac{4 - 2}{2 cdot 1} = frac{2}{2} = 1 in D \\ x_2 = frac{4 + 2}{2 cdot 1} = frac{6}{2} = 3 in D[/latex] [latex]t = x^2 - 4x wedge t = 5 \ x^2 - 4x = 5 \ x^2 - 4x - 5 = 0 \ Delta = (-4)^2 - 4 cdot 1 cdot (-5) = 16 +20= 36; sqrt{Delta} = sqrt{36} = 6 \\ x_1 = frac{4 - 6}{2 cdot 1} = frac{-2}{2} = -1 in D \\ x_2 = frac{4 + 6}{2 cdot 1} = frac{10}{2} = 5 in D[/latex] Stąd: [latex]x in {-1; 1; 3; 5 }[/latex] c) [latex]frac{x^2-6x+3}{4} - frac{5}{x^2-6x+3} = 2[/latex] Zał. [latex]x^2-6x+3 eq 0 \ Delta = (-6)^2 - 4 cdot 1 cdot 3 = 36 - 12= 24; sqrt{Delta} = sqrt{24} = sqrt{4 cdot 6} = 2sqrt{6}\\ x_1 = frac{6 - 2sqrt{6}}{2 cdot 1} = frac{2 cdot (3- sqrt{6})}{2} =3-sqrt{6} approx 0,55 \\ x_2 = frac{6 + 2sqrt{6}}{2 cdot 1} = frac{2 cdot (3+ sqrt{6})}{2} =3+sqrt{6} approx 5,45 \\ D = R setminus {3-sqrt{6}; 3+sqrt{6} }[/latex] [latex]frac{x^2-6x+3}{4} - frac{5}{x^2-6x+3} = 2 \\ t = x^2-6x+3 wedge x^2-6x+3 eq 0 => t eq 0 \\ frac{t}{4} - frac{5}{t} = 2 \\ frac{t}{4} - frac{5}{t} - 2 = 0 \\ frac{t^2}{4t} - frac{20}{4t} - frac{8t}{4t} = 0 \\ frac{t^2 - 20 - 8t}{4t} = 0 / cdot 4t \\ t^2 - 8t - 20 = 0 \ Delta = (-8)^2 - 4 cdot 1 cdot (-20) = 64 + 80 = 144; sqrt{Delta} = sqrt{144} = 12 \\ t_1 = frac{8-12}{2 cdot 1} = frac{-4}{2} = -2 \\ t_2 = frac{8+12}{2 cdot 1} = frac{20}{2} = 10[/latex] [latex]t = x^2 - 6x+3 wedge t = -2 \ x^2 - 6x +3= -2 \ x^2 - 6x +3+2=0 \ x^2 - 6x + 5 = 0 \ Delta = (-6)^2 - 4 cdot 1 cdot 5 = 36 -20= 16; sqrt{Delta} = sqrt{16} = 4 \\ x_1 = frac{6 - 4}{2 cdot 1} = frac{2}{2} = 1 in D \\ x_2 = frac{6 + 4}{2 cdot 1} = frac{10}{2} = 5 in D[/latex] [latex]t = x^2 - 6x+3 wedge t = 10 \ x^2 - 6x +3= 10 \ x^2 - 6x +3-10=0 \ x^2 - 6x -7 = 0 \ Delta = (-6)^2 - 4 cdot 1 cdot (-7) = 36 +28= 64; sqrt{Delta} = sqrt{64} = 8 \\ x_1 = frac{6 - 8}{2 cdot 1} = frac{-2}{2} = -1 in D \\ x_2 = frac{6 + 8}{2 cdot 1} = frac{14}{2} = 7 in D[/latex] Stąd: [latex]x in {-1; 1; 5; 7 }[/latex]
