Chapter 4.7, Problem 32WE

To determine

To find all double roots.

Answer to Problem 32WE

The values of double roots are $2\text{ and }-2$

Explanation of Solution

Given information:

  $x^2\left(x^4+16\right)=8x^4$

Calculation:

Thedouble rootscan be calculated as

  $\begin{array}{l}{x}^2\left(x^4+16\right)=8x^4\\ x^2\left(x^4+16\right)-8x^4=0\left[\text{make one side equal to 0}\right]\\ x^6+16x^2-8x^4=0\\ {\left(x^3\right)}^2-2\left(x^3\right)\left(4x\right)+{\left(4x\right)}^2=0\left[\text{perfect squares trinomial}\right]\\ {\left[\left(x^3\right)-4x\right]}^2=0\end{array}$

Therefore,

  $\begin{array}{l}{\left[\left(x^3\right)-4x\right]}^2=0\left[\text{use the zero-product property}\right]\\ x^3-4x=0\\ x^3=4x\\ x^2=4\\ x=\pm 2\end{array}$

Hence,

The values of double roots are $2\text{ and }-2$