Question

Solve each exponential equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places. Verify your results using a graphing utility.

$2^{2x}+2^{x+2}-12=0$

Short answer

The solution of the given equation is $x=1$.

Step-by-step solution

Step 1. Given information.

The given equation is:

$2^{2x}+2^{x+2}-12=0$

Step 2. Determine the factored form of the equation.

The given equation can be written as:

$\begin{array}{rcl}(2^x{)}^2+2^x·2^2-12& =& 0\\ (2^x{)}^2+4(2^x)-12& =& 0\end{array}$

Let $2^x=u$.

$u^2+4u-12=0$

Factorize the above equation.

$\begin{array}{rcl}{u}^2+6u-2u-12& =& 0\\ u(u+6)-2(u+6)& =& 0\\ (u+6)(u-2)& =& 0\\ (2^x+6)(2^x-2)& =& 0\end{array}$

Step 3. Determine the solution.

Using the zero product property, we get

$\begin{array}{rcl}{2}^x+6& =& 0\text{or}{2}^x-2=0\\ 2^x& =& -6\text{or}{2}^x=2\end{array}$

The equation localid="1648275564805" $2^x=-6$has no solution because localid="1648275578108" $2^x>0$. Now, solve the equation $2^x=2$.

localid="1648275429265" $\begin{array}{rcl}{2}^x& =& 2^1\\ x& =& 1\end{array}$

The solution of the given equation is $x=1$.

Step 4. Verify the result by the graphing utility.

Plot the graph of $y=2^{2x}+2^{x+2}-12$ on a coordinate plane. The x-intercept of the curve is the solution of the given equation.

The graph intersects the x-axis at $x=1$. Hence, the solution is verified.