[latex]a.\ -5 x^{4} +3 x^{3} +14 x^{2} =0\ -x^2(5x^2-3x-14)=0\ -x^2=0 vee 5x^2-3x-14=0\ Delta=b^2-4ac=(-3)^2-4*5*(-14)=289\ x_{1}=frac{-b-sqrt{Delta}}{2a}=frac{3-17}{10}=frac{-14}{10}=-frac{7}{5}\ x_{2}=frac{-b+sqrt{Delta}}{2a}=frac{3+17}{10}=2\ \ x={-frac{7}{5},0,2}\ \ \ b. \ x^{3} +2 x^{2} -16x-32=0\ x^2(x+2)-16(x+2)=0\ (x^2-16)(x+2)=0\ (x-4)(x+4)(x+2)=0\ x-4=0 vee x+4=0 vee x+2=0\ x=4 vee x-4 vee x=-2\ \ x={-4,-2,4}[/latex] [latex]c.\ 2 x^{2} -x+5=0\ Delta=(-1)^2-4*2*5=1-40=-39\ \ [/latex] delta ujemna brak rozwiązań rzeczywistych [latex]d.\ x^{2} +8x+12=0\ Delta=8^2-4*1*12=64-48=16\ x_{1}=frac{-8-4}{2}=-6\ x_{2}=frac{-8+4}{2}=-2\ \ x={-6,-2}[/latex]
a) -5x^4+3x^3+14x^2=-x^2(5x^2-3x-14)=-x^2(5x^2-10x+7x-14)= -x^2[5x(x-2)+7(x-2)]= -x^2(5x+7)(x-2)=0 x=0 lub x= -1,4 lub x=2 b)x^3+2x^2-16x-32=x^2(x+2)-16(x+2)=(x^2-16)(x+2)=(x-4)(x+4)(x+2)=0 x=4 lub x= -4 lub x=-2 c)2x^2-x+5=0 delta=(-1)^2-4*2*5=1-4-= -39 -39<0 (brak pierwiastków) d)x^2+8x+12=x^2-2x-6x+12=x(x-2)-6(x-2)=(x-6)(x-2)=0 x=6 lub x=2
