1. a) [latex]x^2 - frac{x}{2}- frac{1}{2}=0/*2 [/latex] [latex]2x^2 -x-1=0[/latex] Δ=b^2-4ac Δ= [latex](-1)^2-4*2*(-1)= 1+8[/latex] Δ=9 √Δ=3 [latex]x_1= frac{-b-√Δ}{2a} x_2= frac{-b+√Δ}{2a} [/latex] [latex]x_1= frac{-(-1)-3}{2*2} = frac{1-3}{4}=frac{-2}{4} = - frac{1}{2} [/latex] [latex]x_2= frac{-(-1)+3}{2*2} =frac{1+3}{4} = frac{4}{4}=1 [/latex] b) [latex]5x^2-6x+6=0[/latex] Δ=[latex](-6)^2-4*5*6=36-120=-84[/latex] Δ<0 ---brak rozwiązań c) [latex]2x^2-9x-35=0[/latex] Δ= [latex](-9)^2 -4*2*(-35)= 81+280=361[/latex] Δ=361 √Δ=19 [latex]x_1= frac{-(-9)-19}{2*2} = frac{9-19}{4}= frac{-10}{4}=- frac{5}{2}=-2 frac{1}{2} [/latex] [latex]x_2= frac{-(-9)+19}{2*2}= frac{9+19}{4}= frac{28}{4}=7 [/latex] d) [latex]4x^2+12x+9=0[/latex] Δ=[latex]12^2-4*4*9=144-144=0[/latex] Δ=0 ---jedno rozwiązanie [latex]x_1=x_2=- frac{b}{2a} [/latex] [latex]x= frac{-12}{2*4}=- frac{12}{8} =- frac{3}{2} =-1 frac{1}{2} [/latex] 2. a) [latex]2x^2+3=-7x[/latex] [latex]2x^2+7x+3=0[/latex] Δ=[latex]7^2-4*2*3=49-24=25[/latex] Δ=25 √Δ=5 [latex]x_1= frac{-7-5}{2*2}= frac{-12}{4}=-3 [/latex] [latex]x_2= frac{-7+5}{2*2}= frac{-2}{4} = -frac{1}{2} [/latex] b) [latex]16x^2+24 sqrt{2}x=-18 [/latex] [latex]16x^2+24 sqrt{2}x+18=0 [/latex] Δ=[latex](24 sqrt{2})^2-4*16*18=1152-1152=0 [/latex] [latex]x_1=x_2[/latex] [latex]x= frac{-24 sqrt{2} }{2*16}= frac{-24 sqrt{2} }{32} = -frac{3 sqrt{2} }{4} [/latex] c) [latex]11(x^2+5)=x[/latex] [latex]11(x^2+5)-x=0[/latex] [latex]11x^2 -x+55=0[/latex] Δ=[latex](-1)^2-4*11*55=1-2420=2419[/latex] Δ<0 ---brak rozwiązań
