we have the polynomial
[tex]4x^4+2x^2-12=0[/tex]
Change the variable
Let
U=2x^2
substitute
[tex]\begin{gathered} (2x^2)^2+2x^2-12=0 \\ U^2+U-12=0 \end{gathered}[/tex]
Solve for U
using the formula
a=1
b=1
c=-12
substitute
[tex]U=\frac{(-1)\pm\sqrt{1^2-4(1)(-12)}}{2(1)}[/tex][tex]\begin{gathered} U=\frac{-1\pm7}{2} \\ U=3 \\ U=-4 \end{gathered}[/tex]
Solve for x
For U=3
[tex]\begin{gathered} 3=2x^2 \\ x^2=\frac{3}{2} \\ \\ x=\pm\sqrt{\frac{3}{2}}=\pm\frac{\sqrt{6}}{2} \end{gathered}[/tex]
For U=-4
[tex]\begin{gathered} -4=2x^2 \\ x^2=-2 \\ x=\pm i\sqrt{2} \end{gathered}[/tex]
Imaginary roots are
[tex]\begin{gathered} x=i\sqrt{2} \\ x=-i\sqrt{2} \end{gathered}[/tex]
Real Roots are
[tex]\begin{gathered} x=\frac{\sqrt{6}}{2} \\ \\ x=-\frac{\sqrt{6}}{2} \end{gathered}[/tex]
Code Piece is R
