Question

\\4 = \sqrt{3n + 1}\\

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
Isolate the radical and square both sides to solve.

Step 1: Isolate the radical term

To solve for \(n\), we first need to get the square root term by itself on one side of the equation. We do this by subtracting \(1\) from both sides of the equation:

$$4 = \sqrt{3n} + 1$$

$$4 - 1 = \sqrt{3n}$$

$$3 = \sqrt{3n}$$

Step 2: Eliminate the radical by squaring both sides

To undo the square root, we square both sides of the equation:

$$(3)^2 = (\sqrt{3n})^2$$

$$9 = 3n$$

Step 3: Solve for \(n\)

Divide both sides by \(3\) to find the value of \(n\):

$$n = \frac{9}{3}$$

$$n = 3$$

Step 4: Verify the solution

We plug \(n = 3\) back into the original equation to make sure it works:

$$4 = \sqrt{3(3)} + 1$$

$$4 = \sqrt{9} + 1$$

$$4 = 3 + 1$$

$$4 = 4$$

The solution is correct.

Answer:

\(n = 3\)