Chapter 7.4, Problem 8WE

(a)

To determine

Solve given equation.

Answer to Problem 8WE

  $x=\pm 2i\text{ or }x=\pm \frac{\sqrt{2}}{2}$

Explanation of Solution

Given:

The equation is :

  $2x^4+7x^2-4=0$

Calculation:

Substitute

  $x^2=z$ in the equation $2x^4+7x^2-4=0$

  $\begin{array}{l}\Rightarrow 2z^2+7z-4=0\\ \Rightarrow 2z^2+8z-z-4=\mathrm{0........}\left[\begin{array}{l}\text{factor the middle term such that the sum of the factors is the middle term and }\\ \text{the product of the factors is the product of first and the last term}\text{.}\end{array}\right]\\ \Rightarrow 2z(z+4)-1(z+4)=\mathrm{0...}[\text{distributive property]}\\ \Rightarrow \left(z+4\right)\left(2z-1\right)=\mathrm{0..........}[\text{distributive property]}\\ \Rightarrow \left(z+4\right)=0\text{ or }\left(2z-1\right)=\mathrm{0...}[ab=0\iff a=0\text{ or b=0]}\\ \Rightarrow z=-4\text{ or z}=\frac{1}{2}\end{array}$ Substitute

  $\begin{array}{l}z=x^2\\ \Rightarrow x^2=-4\text{ or }{x}^2=\frac{1}{2}\\ \Rightarrow x=\pm \sqrt{-4}\text{ or }x=\pm \sqrt{\frac{1}{2}}\mathrm{.......}[\text{take square root each side]}\\ \Rightarrow x=\pm 2i\text{ or }x=\pm \frac{\sqrt{2}}{2}\mathrm{...............}[\sqrt{-1}=i;rationalize]\end{array}$

(b)

To determine

Solve given equation.

Answer to Problem 8WE

  $x=1\pm 2i\text{ or }x=1\pm \frac{\sqrt{2}}{2}$

Explanation of Solution

Given:

The equation is :

  $2{(x-1)}^4+7{(x-1)}^2-4=0$

Calculation:

Substitute

  ${(x-1)}^2=z$ in the equation $2{(x-1)}^4+7{(x-1)}^2-4=0$

  $\begin{array}{l}\Rightarrow 2z^2+7z-4=0\\ \Rightarrow 2z^2+8z-z-4=\mathrm{0........}\left[\begin{array}{l}\text{factor the middle term such that the sum of the factors is the middle term and }\\ \text{the product of the factors is the product of first and the last term}\text{.}\end{array}\right]\\ \Rightarrow 2z(z+4)-1(z+4)=\mathrm{0...}[\text{distributive property]}\\ \Rightarrow \left(z+4\right)\left(2z-1\right)=\mathrm{0..........}[\text{distributive property]}\\ \Rightarrow \left(z+4\right)=0\text{ or }\left(2z-1\right)=\mathrm{0...}[ab=0\iff a=0\text{ or b=0]}\\ \Rightarrow z=-4\text{ or z}=\frac{1}{2}\end{array}$ Substitute

  $\begin{array}{l}$ z={(x-1)}^2$$ \Rightarrow {(x-1)}^2=-4\text{ or }{(x-1)}^2=\frac{1}{2}$\Rightarrow x-1=\pm \sqrt{-4}\text{ or }x-1=\pm \sqrt{\frac{1}{2}}\mathrm{.......}[\text{take square root each side]}\\ \Rightarrow x-1=\pm 2i\text{ or }x-1=\pm \frac{\sqrt{2}}{2}\mathrm{...............}[\sqrt{-1}=i;rationalize]\\ \Rightarrow x=1\pm 2i\text{ or }x=1\pm \frac{\sqrt{2}}{2}\mathrm{.............}[\text{add 1 each side]}\end{array}$