The solutions of the equation are [tex]x=\pm i, x=\pm \sqrt{5}i[/tex]
Explanation:
Given that the equation is [tex]x^{4}+6 x^{2}+5=0[/tex]
We need to determine the solutions of the equation.
Let us substitute [tex]x^{2} =u[/tex] and [tex]x^4=u^2[/tex]
Thus, the equation becomes,
[tex]u^{2}+6 u+5=0[/tex]
Factoring the equation, we get;
[tex](u+1)(u+5)=0[/tex]
[tex]u=-1, u=-5[/tex]
Substituting back [tex]x^{2} =u[/tex] and solve for x.
First, we shall substitute [tex]u=-1[/tex]
Thus, we get;
[tex]x^{2} =-1[/tex]
[tex]x=\sqrt{-1}[/tex]
[tex]x=\pm i[/tex]
Similarly, substituting [tex]u=-5[/tex], we get;
[tex]x^{2} =-5[/tex]
[tex]x=\sqrt{-5}[/tex]
[tex]x=\pm \sqrt{5}i[/tex]
Thus, the solutions of the equation are [tex]x=\pm i, x=\pm \sqrt{5}i[/tex]
