a]
[tex]2x^2-9x-35=0\\\\a=2, \ b=-9, \ c=-35\\\\\Delta=(-9)^2-4\cdot2\cdot(-35)=81+280=361\\\\\sqrt{\Delta}=\sqrt{361}=19\\\\x_1=\frac{-(-9)-19}{2\cdot2}=\frac{-10}{4}=-2,5\\\\x_2=\frac{-(-9)+19}{2\cdot2}=\frac{28}{4}=7[/tex]
b]
[tex]4x^2-13x+3=0\\\\a=4, \ b=-13, \ c=3\\\\\Delta=(-13)^2-4\cdot4\cdot3=169-48=121\\\\\sqrt{\Delta}=\sqrt{121}=11\\\\x_1=\frac{-(-13)-11}{2\cdot4}=\frac{2}{8}=\frac{1}{4}\\\\x_2=\frac{-(-13)+11}{2\cdot4}=\frac{24}{8}=3[/tex]
c]
[tex]-6x^2+13x+5=0 \ \ |\cdot(-1)\\\\6x^2-13x-5=0\\\\a=6, \ b=-13, \ c=5\\\\\Delta=(-13)^2-4\cdot6\cdot5=169-120=49\\\\\sqrt{\Delta}=\sqrt{49}=7\\\\x_1=\frac{-(-13)-7}{2\cdot6}=\frac{6}{12}=\frac{1}{2}\\\\x_2=\frac{-(-13)+7}{2\cdot6}=\frac{20}{12}=1\frac{8}{12}=1\frac{2}{3}[/tex]
d]
[tex]5x^2-6x+6=0\\\\a=5, \ b=-6, \ c=6\\\\\Delta=(-6)^2-4\cdot5\cdot6=36-120=-84<0[/tex]
Brak rozwiązań
e]
[tex]-2x^2+5x-3=0 \ \ |\cdot(-1)\\\\2x^2-5x+3=0\\\\a=2, \ b=-5, \ c=3\\\\\Delta=(-5)^2-4\cdot2\cdot3=25-24=1\\\\\sqrt{\Delta}=\sqrt1=1\\\\x_1=\frac{-(-5)-1}{2\cdot2}=\frac{4}{4}=1\\\\x_2=\frac{-(-5)+1}{2\cdot2}=\frac{6}{4}=1,5[/tex]
f]
[tex]4x^2+12x+9=0\\\\a=4, \ b=12, \ c=9\\\\\Delta=12^2-4\cdot4\cdot9=144-144=0\\\\x_0=\frac{-12}{2\cdot4}=\frac{-12}{8}=-1,5[/tex]
g]
[tex]\frac{1}{2}x^2+x+1=0 \ \ |\cdot2\\\\x^2+2x+2=0\\\\a=1, \ b=2, \ c=2\\\\\Delta=2^2-4\cdot1\cdot2=4-8=-4<0[/tex]
Brak rozwiązań
h]
[tex]x^2-\frac{x}{2}-\frac{1}{2}=0 \ \ |\cdot2\\\\2x^2-x-1=0\\\\a=2, \ b=-1, \ c=-1\\\\\Delta=(-1)^2-4\cdot2\cdot(-1)=1+8=9\\\\\sqrt{\Delta}=\sqrt9=3\\\\x_1=\frac{-(-1)-3}{2\cdot2}=\frac{-2}{4}=-\frac{1}{2}\\\\x_2=\frac{-(-1)+3}{2\cdot2}=\frac{4}{4}=1[/tex]
i]
[tex]\frac{1}{4}x^2-\frac{x}{3}+\frac{1}{9}=0 \ \ |\cdot36\\\\9x^2-12x+4=0\\\\a=9, \ b=-12, \ c=4\\\\\Delta=(-12)^2-4\cdot9\cdot4=144-144=0\\\\x_0=\frac{-(-12)}{2\cdot9}=\frac{12}{18}=\frac{2}{3}[/tex]
