Question

Find all complex solutions of the equation x^4 + x^2 - 6 = 0

Answer 1

`x^4+x^2-6=0\\ x^4+3x^2-2x^2-6=0\\ x^2(x^2+3)-2(x^2+3)=0\\ (x^2-2)(x^2+3)=0\\ x^2-2=0 \vee x^2+3=0\\ x^2=2 \vee x^2=-3\\ x=-\sqrt2 \vee x=\sqrt2 \vee x=-i\sqrt3 \vee x=i\sqrt3`

Answer 2

`x^4+x^2-6=0 \\ (x^2)^2+x^2-6=0 \\ \\ a=1 \\ b=1 \\ c=-6 \\ b^2-4ac=1^2-4 * 1 * (-6)=1+24=25 \\ \\ x^2=(-b \pm √(b^2-4ac))/(2a)=(-1 \pm √(25))/(2 * 1)=(-1 \pm 5)/(2) \\ x^2=(-1-5)/(2) \ \lor \ x^2=(-1+5)/(2) \\ x^2=(-6)/(2) \ \lor \ x^2=(4)/(2) \\ x^2=-3 \ \lor \ x^2=2 \\ x=\pm √(-3) \ \lor \ x=\pm √(2) \\ \boxed{x=-i√(3) \hbox{ or } x=i√(3) \hbox{ or } x=-√(2) \hbox{ or } x=√(2)}`