Chapter 6.5, Problem 42PPS

To determine

To solve: the given equation.

Answer to Problem 42PPS

  $x=\pm i,\pm i\sqrt{5}$ .

Explanation of Solution

Given:

The given equation is $x^4+6x^2+5=0$ .

Calculation:

The given equation is $x^4+6x^2+5=0$ .

In order to solve the equation, factor the equation. Therefore,

  $\begin{array}{l}{x}^4+6x^2+5=0\left\{\text{given equation}\right\}\\ x^4+x^2+5x^2+5=0\left\{\text{express in simplified form}\right\}\\ x^2\left(x^2+1\right)+5\left(x^2+1\right)=0\left\{\text{taking }\left(x^2+1\right)\text{as common}\right\}\\ \left(x^2+5\right)\left(x^2+1\right)=0\left\{\text{use distributive property of product}\right\}\end{array}$

Either $\left(x^2+5\right)=0$ or $\left(x^2+1\right)=0$

when $\left(x^2+1\right)=0$ , then $x^2=-1\Rightarrow x=\pm \sqrt{-1}=\sqrt{i^2\times 1}=\pm i$

when $\left(x^2+5\right)=0$ , then $x^2=-5\Rightarrow x=\pm \sqrt{-5}=\sqrt{i^2\times 5}=\pm \sqrt{5}i$ :

Hence, the solution of the given equation is $x=\pm i,\pm i\sqrt{5}$ .