Chapter 3, Problem 44CLT

To determine

To solve the exponential equation:

  $2x^2{e}^{2x}-2x{e}^{2x}=0$

And to verify the result graphically.

Answer to Problem 44CLT

105.966

Explanation of Solution

Given:

  $2x^2{e}^{2x}-2x{e}^{2x}=0$

Concept Used:

For any x : $e^x\ne 0$

  $u=e^x\iff \ln \left(u\right)=x$

Calculation:

To solve the exponential equation:

  $2x^2{e}^{2x}-2x{e}^{2x}=0$

First, simplify the equation as,

  $\begin{array}{c}2x^2{e}^{2x}-2x{e}^{2x}=0\\ \Rightarrow 2x{e}^{2x}\left(x-1\right)=0\end{array}$

Now since $e^x\ne 0$ for any x.

Thus,

  $\begin{array}{c}2x{e}^{2x}\left(x-1\right)=0\\ \Rightarrow x\left(x-1\right)=0\\ \Rightarrow x=0,\text{and }x=1\end{array}$

Thus, solution of given equation are 0 and 1.

Now, to verify this result graphically plot the graph of function $2x^2{e}^{2x}-2x{e}^{2x}$ and then check whether at x equals to 0, and 1 value of the function is 0, that is, check whether the points $\left(0,0\right)\text{ and }\left(1,0\right)$ lies on the curve of this graph.

  

Since, the points $\left(0,0\right)\text{ and }\left(1,0\right)$ lies on the graph of the function $2x^2{e}^{2x}-2x{e}^{2x}$ , thus the result is verified.