[latex]a)\ x^4 - 6x^3 + 5x^2=0\ \x^2(x^2-6x+5)=0\ \x^2=0vee x^2-6x+5 =0\\Delta = b^{2}-4ac = (-6)^{2}-4*1*5=36-20=16 \ \sqrt{Delta }=sqrt{16}=4[/latex] [latex]x_{1}=frac{-b-sqrt{Delta }}{2a} =frac{6-4}{2}=frac{-6}{2}=-3 \ \x_{2}=frac{-b+sqrt{Delta }}{2a} =frac{6+4}{2}=frac{10}{2} = 5 \ \x=0vee x=-3 vee x=5[/latex] [latex]b)\x^4 +6x^3 +8x^2=0\ \x^2(x^2+6x+8)=0\ \x^2=0vee x^2+6x+8 =0\\Delta = b^{2}-4ac = 6^{2}-4*1*8=36-32=4\ \sqrt{Delta }=sqrt{4}=2[/latex] [latex]x_{1}=frac{-b-sqrt{Delta }}{2a} =frac{-6-2}{2}=frac{-8}{2}=-4\ \x_{2}=frac{-b+sqrt{Delta }}{2a} =frac{-6+2}{2}=frac{-4}{2} = -2 \ \x=0vee x=-4 vee x=-2[/latex] [latex]c)\x^4 -2x^3 - 24x^2 =0\ \x^2(x^2-2x-24)=0\ \x^2=0vee x^2-2x-24 =0\\Delta = b^{2}-4ac = (-2)^{2}-4*1*(-24)=4+96=100\ \sqrt{Delta }=sqrt{100}=10[/latex] [latex]x_{1}=frac{-b-sqrt{Delta }}{2a} =frac{2-10}{2}=frac{-8}{2}=-4\ \x_{2}=frac{-b+sqrt{Delta }}{2a} =frac{2+10}{2}=frac{12}{2} =6 \ \x=6vee x=-4 vee x=0[/latex] [latex]d)\x^4 -12x^3 + 36x^2 =0\ \x^2(x^2-12x+36)\ \x^2=0vee x^2-12+36 =0\\Delta = b^{2}-4ac = (-12)^{2}-4*1*36=144-144=0\ \x_{o}=frac{-b}{2a}=frac{12}{2}=6\ \x=0vee x=6[/latex] [latex]e)\4x^4 +4x^3 -3x^2 =0\ \x^2(4x^2+4x-3)=0\ \x^2=0vee 4x^2+4x-3=0\\Delta = b^{2}-4ac = 4^{2}-4*4*(-3)=16+48=64\ \sqrt{Delta} =sqrt{64}=8[/latex] [latex]x_{1}=frac{-b-sqrt{Delta }}{2a} =frac{-4-8}{2*4}=frac{-12}{8}=-frac{3}{2}\ \x_{2}=frac{-b+sqrt{Delta }}{2a} =frac{-4+8}{2*4}=frac{4}{8} =frac{1}{2} \ \x=-frac{3}{2}vee x=frac{1}{2} vee x=0[/latex] [latex]f)\12x^4 +5x^3 -2x^2 =0\ \x^2(12x^2+5x-2)=0\ \x^2=0vee 12x^2+5x-2\\Delta = b^{2}-4ac = 5^{2}-4*12*(-2)=25+96=121\ \sqrt{Delta} =sqrt{121}=11[/latex] [latex]x_{1}=frac{-b-sqrt{Delta }}{2a} =frac{-5-11}{2*12}=frac{-16}{24}=-frac{2}{3}\ \x_{2}=frac{-b+sqrt{Delta }}{2a} =frac{-5+11}{2*12}=frac{6}{24} =frac{1}{4} \ \x=-frac{2}{3}vee x=frac{1}{4} vee x=0[/latex]
