[tex]a)\\\\ f(x) = x^2 + 2x-3 \\\\a=1,\ \ b=2,\ \ c=-3\\\\\Delta =b^2-4ac=2^2-4*1*(-3)=4+12=16\\\\\sqrt{\Delta }=\sqrt{16}=4\\\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a}=\frac{-2-4}{2*1}=\frac{-6}{2}=-3\\\\x_{2}=\frac{-b+\sqrt{\Delta }}{2a}=\frac{-2+4}{2*1}=\frac{2}{2}=1[/tex]
[tex]postac\ iloczynowa :\\\\y=a(x-x_{1})(x-x_{2})\\\\y=(x+3)(x-1)[/tex]
[tex]b)\\\\ f(x) = x^2-10x + 25 \\\\a=-10,\ \ b=2,\ \ c=25\\\\\Delta =b^2-4ac= (-10)^2-4*1*25=100-100=0 \\\\ x_{o}=\frac{-b }{2a}=\frac{10}{2*1}= 5[/tex]
[tex]postac\ iloczynowa :\\\\y= (x- 5)(x- 5)=(x-5)^2[/tex]
[tex]c )\\\\ f(x) = x^2 +x-6 \\\\a=1,\ \ b=1,\ \ c=-6\\\\\Delta =b^2-4ac= 1^2-4*1*(-6)=1+24=25\\\\\sqrt{\Delta }=\sqrt{25}=5\\\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a}=\frac{-1-5}{2*1}=\frac{-6}{2}=-3\\\\x_{2}=\frac{-b+\sqrt{\Delta }}{2a}=\frac{-1+5}{2*1}=\frac{4}{2}=2[/tex]
[tex]c )\\\\ f(x) = x^2 +x-6 \\\\a=1,\ \ b=1,\ \ c=-6\\\\\Delta =b^2-4ac= 1^2-4*1*(-6)=1+24=25\\\\\sqrt{\Delta }=\sqrt{25}=5\\\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a}=\frac{-1-5}{2*1}=\frac{-6}{2}=-3\\\\x_{2}=\frac{-b+\sqrt{\Delta }}{2a}=\frac{-1+5}{2*1}=\frac{4}{2}=2[/tex]
[tex]d)\\\\ f(x) = -2x^2 -4x-2 \\\\a=-2,\ \ b=-4,\ \ c=-2\\\\ \Delta =b^2-4ac= (-4)^2-4*(-2)*(-2)=16-16=0 \\\\ x_{o}=\frac{-b }{2a}=\frac{ 4}{2* (-2)}= \frac{4}{-4}=-1\\\\postac\ iloczynowa :\\\\y= -2(x+1)(x+1)=-2(x+1)^2[/tex]
[tex]e)\\\\ f(x) = -3x^2+4x-5 \\\\a=-3,\ \ b= 4,\ \ c=-5\\\\ \Delta =b^2-4ac= 4^2-4*(-3)*(-5)=16-60=-44\\\\\Delta <0\\\\-\ brak\ pierwiastkow\\\\-\ brak\ postaci\ iloczynowej[/tex]
