Question

Solve the equation. Check for extraneous solutions. $$\sqrt{\frac{1}{4} x-4}-3=5$$

Short answer

The solution to the equation is \(x = 272\), and there are no extraneous solutions.

Step-by-step solution

Isolate the square root term on one side

Begin by isolating the square root term on one side of the equation by adding 3 to both sides to get rid of -3: \(\sqrt{\frac{1}{4} x-4} = 5 + 3\), which simplifies to \(\sqrt{\frac{1}{4} x-4} = 8\)

Eliminate the square root

Next, square both sides of the equation to eliminate the square root: \((\sqrt{\frac{1}{4} x-4})^2 = 8^2\), which simplifies to \(\frac{1}{4}x - 4 = 64\)

Simplify the equation

Simplify the equation by adding 4 to both sides: \(\frac{1}{4}x = 64 + 4\), which simplifies to \(\frac{1}{4}x = 68\)

Solve for x

Solve for x by multiplying both sides of the equation by 4: \(x = 68 * 4\), which simplifies to \(x = 272\)

Check for extraneous solutions

Substitute 272 for x in the original equation to verify it as a valid solution: \(\sqrt{\frac{1}{4} * 272-4}-3 = 5\). This simplifies to \(\sqrt{68-4}-3 = 5\), then \(\sqrt{64} - 3 = 5\), 8-3 = 5, which confirms that x = 272 is not an extraneous solution.