Question

The following equation has four solutions: x^4+6x^2=-8The two imaginary solutions with rational coefficients are +__i and two imaginary solutions with irrational coefficients are +I✔️(__).

Answer 1

Given

The equation,

$x^4+6x^2=-8$

To find:

The roots of the given equation.

Explanation:

It is given that,

$x^4+6x^2=-8$

That implies,

$\begin{gathered} x^4+6x^2=-8 \\ x^4+6x^2+8=0 \end{gathered}$

Put x²=y.

Then,

$\begin{gathered} y^2+6y+8=0 \\ y=\frac{-6\pm\sqrt{36-32}}{2} \\ y=\frac{-6\pm\sqrt{4}}{2} \\ y=\frac{-6\pm2}{2} \\ y=\frac{-6+2}{2},\text{ }y=\frac{-6-2}{2} \\ y=\frac{-4}{2},\text{ }y=\frac{-8}{2} \\ y=-2,\text{ }y=-4 \end{gathered}$

Therefore,x

$\begin{gathered} x^2=-2,\text{ }x^2=-4 \\ x=\pm\sqrt{-2},\text{ }x=\pm\sqrt{-4} \\ x=\pm i\sqrt{2},\text{ }x=\pm2i \end{gathered}$

Hence, the two imaginary ssolutions with rational coefficients are ±2i, and the two imaginary solutions with irrational coefficients are ±i√(2).