Question

In the following exercises, solve each equation.

$2^{x^2+2x}=\frac{1}{2}$

Short answer

The answer is$x=-1$

Step-by-step solution

Step 1. Given

The equation is$2^{x^2+2x}=\frac{1}{2}$

Step 2. Solve the equation

$$\begin{gathered} 2^{x^2+2x}=\frac{1}{2} \\ {2}^{x^2+2x}=2^{-1} \end{gathered}$$

Bases are same so the exponents should be equal.

$$\begin{gathered} x^2+2x=-1 \\ {x}^2+2x+1=0 \\ {\left(x+1\right)}^2=0 \\ x+1=0 \\ x=-1 \end{gathered}$$

Step 3. Check the answer

$$\begin{gathered} 2^{x^2+2x}=\frac{1}{2} \\ {2}^{{\left(-1\right)}^2+2\left(-1\right)}=\frac{1}{2} \\ {2}^{1-2}=\frac{1}{2} \\ {2}^{-1}=\frac{1}{2} \\ \frac{1}{2}=\frac{1}{2} \end{gathered}$$

True