Question

Solve:$2\sqrt{4a+2}-16=16$.

Short answer

The solution is$a=\frac{127}{2}$.

Step-by-step solution

Step 1. Given Information 

We are given the quadratic equation $2\sqrt{4a+2}-16=16$and we need to solve it.

Step 2. Finding the solution  

Isolating the radical on one side of the equation:

$$\begin{gathered} 2\sqrt{4a+2}-16=16 \\ \Rightarrow 2\sqrt{4a+2}=16+16 \\ \Rightarrow 2\sqrt{4a+2}=32 \end{gathered}$$

Squaring both sides of the equation, we get:

$$\begin{gathered} {\left(2\sqrt{4a+2}\right)}^2={\left(32\right)}^2 \\ \Rightarrow 4\left(4a+2\right)=1024 \\ \Rightarrow 16a+8=1024 \\ \Rightarrow 16a=1024-8 \\ \Rightarrow 16a=1016 \\ \Rightarrow a=\frac{1016}{16} \\ \Rightarrow a=\frac{127}{2} \end{gathered}$$

Step 3. Checking the answer  

$$\begin{gathered} a=\frac{127}{2}, \\ 2\sqrt{4a+2}-16=16 \\ \Rightarrow 2\sqrt{4\times \frac{127}{2}+2}-16=16 \\ \Rightarrow 2\sqrt{254+2}-16=16 \\ \Rightarrow 2\sqrt{256}-16=16 \\ \Rightarrow 2\times 16-16=16 \\ \Rightarrow 32-16=16 \\ \Rightarrow 16=16 \end{gathered}$$