[latex]a) x^{2}+8x+16=0\ x^{2}+2*x*4+4^{2}=0\ (x+4)^{2}=0\ x+4=0\ x=-4\ -------------\ (a+b)^{2}=a^{2}+2ab+b^{2}[/latex] [x=-4 - pierwiastek dwukrotny] ================================== [latex]b) x^{2}-x+frac{1}{4}=0\ x^{2}-2*x*frac{1}{2}+(frac{1}{2})^{2}=0\ (x-frac{1}{2})^{2}=0\ x-frac{1}{2}=0\ x=frac{1}{2}\ --------------\ (a-b)^{2}=a^{2}-2ab+b^{2}[/latex] [x=1/2 - pierwiastek dwukrotny] ================================== [latex]c) x^{2}+x-1=0\ d=b^{2}-4ac=1^{2}-4*1*(-1)=1+4=5\ sqrt{d}=sqrt{5}\ x_{1}=frac{-b-sqrt{d}}{2a}=frac{-1-sqrt{5}}{2}\ x_{2}=frac{-b+sqrt{d}}{2a}=frac{-1+sqrt{5}}{2}[/latex] ================================== [latex]d) -x^{2}+frac{1}{2}x+3=0 |*2\ -2x^{2}+x+6=0\ d=b^{2}-4ac=1^{2}-4*(-2)*6=1+48=49\ sqrt{d}=7\ x_{1}=frac{-b-sqrt{d}}{2a}=frac{-1-7}{-4}=frac{-8}{-4}=2\ x_{1}=frac{-b+sqrt{d}}{2a}=frac{-1+7}{-4}=frac{6}{-4}=-frac{3}{2}[/latex]
