Q. 80 - Precalculus Enhanced with Graphing Utilities

Question

Solve given equation and verify your result using graphing utility. 

${\left(e^4\right)}^x.e^{x2}=e^{12}$

Step-by-Step Solution

Answer

The solution set is $\{2 , -6\}$ and it is verified through graph. 

Step 1. Given information

The given equation is :  

${\left(e^4\right)}^x.e^{x2}=e^{12}$

We have to solve for x and verify using graphing utility.  

Step 2. Get a single expression with base e on both sides.

Using law of exponents  , get a single expression with base e on left side . 

${\left(e^4\right)}^x.e^{x2}= e^{4x}.e^{x^2}=e^{4x+x^2}$

Step 3. Comparing the expressions having same bases and solving for x .

Since bases are same , therefore the exponential powers should also be the same.  

$$\begin{gathered} 4x+x^2=12 \\ {x}^2+4x-12=0 \end{gathered}$$

Factorise above expression 

$$\begin{gathered} (x+6)(x-2)=0 \\ x+6=0 , x-2=0 \\ x=-6 , x=2 \end{gathered}$$

Step 4. Verification using graphing utility

Graph

$$\begin{gathered} Y_1=e^{4x + x^2} \\ {Y}_2=e^{12} \end{gathered}$$

Determine the point of intersection.  

The graph intersects at $x=2, x=-6$

So the solution set is $\{2, -6\}$