[tex]Sa\ to\ rowniania\ kwadratowe :\\\\-\ obliczam\ delte\ a\ potem\ pierwiastki\\\\ a)\\\\y=2x^2-2x-1\\\\a=2,\ \ b=-2,\ \ c=-1\\\\\Delta=b^2-4ac=(-2)^2-4*2*(-1)=4+8=12\\\\\sqrt{\Delta}=\sqrt{12}=\sqrt{4*3}=2\sqrt{3}\\\\x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{- (-2)- 2\sqrt{3}}{2 *2}=\frac{2- 2\sqrt{3}}{2 *2} =\frac{\not{2}^1(1- \sqrt{3})}{\not{2}^1 *2} =\frac{ 1- \sqrt{3} }{ 2}[/tex]
[tex]x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{- (-2)+ 2\sqrt{3}}{2 *2}=\frac{2+ 2\sqrt{3}}{2 *2} =\frac{\not{2}^1(1+ \sqrt{3})}{\not{2}^1 *2} =\frac{ 1+ \sqrt{3} }{ 2}[/tex]
[tex]b)\\\\y=4x^2-2x-1\\\\a=4,\ \ b=-2,\ \ c=-1\\\\\Delta=b^2-4ac=(-2)^2-4*4*(-1)=4+16=20\\\\\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=2\sqrt{5}\\\\x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{- (-2)- 2\sqrt{5}}{2 *4}=\frac{2- 2\sqrt{5}}{2 *4} =\frac{\not{2}^1(1- \sqrt{5})}{\not{2}^1 *4} =\frac{ 1- \sqrt{5} }{4}\\\\x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{- (-2)+ 2\sqrt{5}}{2 *4}=\frac{2+ 2\sqrt{5}}{2 *4} =\frac{\not{2}^1(1+ \sqrt{5})}{\not{2}^1 *4} =\frac{ 1+ \sqrt{5} }{4}[/tex]
[tex]c)\\\\y= x^2-4x-4\\\\a=1,\ \ b=-4,\ \ c=-4\\\\\Delta=b^2-4ac=(-4)^2-4*1*(-4)=16+16=32\\\\\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=4\sqrt{2}\\\\x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{- (-4)- 4\sqrt{2}}{2 *1}=\frac{4- 4\sqrt{2}}{2 } =\frac{\not{2}^1(2- 2\sqrt{2})}{\not{2}^1 } = 2-2\sqrt{2}\\\\x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{- (-4)+ 4\sqrt{2}}{2 *1}=\frac{4+ 4\sqrt{2}}{2 } =\frac{\not{2}^1(2+ 2\sqrt{2})}{\not{2}^1 } = 2+2\sqrt{2}[/tex]
