Question

Solve the equation. Check for extraneous solutions. $$\sqrt{5 x+1}+8=12$$

Short answer

The solution to the equation \( \sqrt{5 x+1}+8=12 \) is \( x = 3 \)

Step-by-step solution

Isolate the square root

First, isolate the square root on one side of the equation by subtracting 8 from both sides of the equation. This gives the new equation: \( \sqrt{5x+1} = 12 - 8 \), which simplifies to \( \sqrt{5x+1} = 4 \).

Remove the square root

Now, eliminate the square root by squaring both sides of the equation. This gives \( (\sqrt{5x+1})^2 = 4^2 \), which simplifies to \( 5x + 1 = 16 \).

Solve for x

Then, solving the equation for x involves subtracting 1 from both sides of the equation: \( 5x = 16 - 1 \), which simplifies to \( 5x = 15 \). Lastly, divide both sides of the equation by 5 to solve for x: \( x = 15 / 5 \), which gives that \( x = 3 \).

Check for an extraneous solution

Check the tentative solution \( x = 3 \) in the original equation \( \sqrt{5x+1}+8=12 \) to verify that it works. Substitute x with 3 in the equation: \( \sqrt{5(3)+1}+8=12 \), this simplifies to \( \sqrt{16}+8=12 \), which gives \( 4+8=12 \). So, this verifies that 12 equals 12, hence, x = 3 is not an extraneous solution and is indeed the solution to the equation.