Answer:
[tex]x=i,\:x=-i,\:x=3i,\:x=-3i[/tex]
Step-by-step explanation:
[tex]0=x^4+10x^2+9[/tex]
[tex]x^4+10x^2+9=0[/tex]
[tex]u^2+10u+9=0[/tex]; [tex]u=x^2, u^2=x^4[/tex]
[tex]u^2+10u+9=0[/tex]
Use quadratic equation:
[tex]\frac{-10\pm \sqrt{10^2-4\cdot \:1\cdot \:9}}{2\cdot \:1}[/tex]
[tex]\frac{-10+\sqrt{10^2-4\cdot \:1\cdot \:9}}{2\cdot \:1}[/tex]
[tex]=\frac{-10+\sqrt{64}}{2\cdot \:1}[/tex]
[tex]=\frac{-10+8}{2}[/tex]
[tex]=-\frac{2}{2}[/tex]
[tex]=1[/tex]
[tex]\frac{-10-\sqrt{10^2-4\cdot \:1\cdot \:9}}{2\cdot \:1}[/tex]
[tex]\frac{-10-\sqrt{64}}{2\cdot \:1}[/tex]
[tex]=\frac{-10-\sqrt{64}}{2}[/tex]
[tex]=\frac{-18}{2}[/tex]
[tex]=-9[/tex]
[tex]u=-1, u=-9[/tex]
Substitute back in:
[tex]x^2=-1[/tex]
[tex]x=\sqrt{-1},\:x=-\sqrt{-1}[/tex]
[tex]x=i,\:x=-i[/tex]
[tex]x^2=-9[/tex]
[tex]x=\sqrt{-9},\:x=-\sqrt{-9}[/tex]
[tex]\sqrt{-9}=3i[/tex], [tex]-\sqrt{-9}=-3i[/tex]
Therefore the solutions are:
[tex]x=i,\:x=-i,\:x=3i,\:x=-3i[/tex]
