Chapter 6.5, Problem 63PPS

To determine

To solve: the given equation.

Answer to Problem 63PPS

  $x=\pm 3,\pm i\sqrt{\frac{3}{5}}$ .

Explanation of Solution

Given:

The given equation is $3x^4-22x^2-45=0$ .

Calculation:

The given equation is $3x^4-22x^2-45=0$ .

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

  $\begin{array}{l}3x^4-22x^2-45=0\left\{\text{given equation}\right\}\\ 3x^4+5x^2-27x^2-45=0\left\{\text{express in simplified form}\right\}\\ x^2\left(3x^2+5\right)-9\left(3x^2+5\right)=0\left\{\text{taking }\left(3x^2+5\right)\text{as common}\right\}\\ \left(x^2-9\right)\left(3x^2+5\right)=0\left\{\text{use distributive property of product}\right\}\end{array}$

Either $\left(x^2-9\right)=0$ or $\left(3x^2+5\right)=0$

when $\left(x^2-9\right)=0$ , then $x^2=9\Rightarrow x=\pm 3$

when $\left(3x^2+5\right)=0$ , then $3x^2=-5\Rightarrow x^2=-\frac{3}{5}\Rightarrow x=\pm \sqrt{-1\times \frac{3}{5}}=\pm i\sqrt{\frac{3}{5}}$

Hence, the solution of the given equation is $x=\pm 3,\pm i\sqrt{\frac{3}{5}}$ .