Problem 16 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{5 n+1}-6=-4\)

Step-by-Step Solution

Answer

The solution is \(n = \frac{3}{5}\).

Step 1: Isolate the Square Root

First, add 6 to both sides of the equation to isolate the square root term. This gives us:\[\sqrt{5n + 1} - 6 + 6 = -4 + 6\]Simplifying both sides gives:\[\sqrt{5n + 1} = 2\]

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation:\[(\sqrt{5n + 1})^2 = 2^2\]This simplifies to:\[5n + 1 = 4\]

Step 3: Solve for n

Subtract 1 from both sides to solve for \(n\):\[5n + 1 - 1 = 4 - 1\]This gives:\[5n = 3\]Now, divide both sides by 5 to solve for \(n\):\[n = \frac{3}{5}\]

Step 4: Verify the Solution

Substitute \(n = \frac{3}{5}\) back into the original equation to verify the solution:\[\sqrt{5 \left(\frac{3}{5}\right) + 1} - 6 = -4\]Calculating inside the square root:\[\sqrt{3 + 1} = \sqrt{4}\]Since \(\sqrt{4} = 2\), substitute back:\[2 - 6 = -4\]Since both sides are equal, \(n = \frac{3}{5}\) is a valid solution.

Key Concepts

Isolating the Square Root

When solving square root equations, the first step is to isolate the term that contains the square root. Think of this as moving the square root part to one side of the equation, all by itself. This step is crucial because it sets the stage for effectively eliminating the square root, making the equation easier to solve. In the given problem, \(\sqrt{5n+1} - 6 = -4\), we start by getting rid of the \(-6\) on the left side by adding 6 to both sides of the equation. This changes the equation to:

• \(\sqrt{5n + 1} = 2\)

A quick recap: isolating the square root means performing basic arithmetic operations to have only the square root on one side of the equation. This simplification is vital for the upcoming steps.

Verify the Solution

Once you think you have the solution to a square root equation, it's important to verify that solution. Verification helps confirm that our calculated value of \(n\) satisfies the original equation. To verify, substitute the solution back into the initial equation. In our example, we found that \(n = \frac{3}{5}\). We put this value in the starting equation:

• Check: \(\sqrt{5(\frac{3}{5}) + 1} - 6 = -4\)

Evaluate inside the square root first:

• \(\sqrt{3 + 1} = \sqrt{4}\)

Since \(\sqrt{4} = 2\), substitute it back:

• \(2 - 6 = -4\)

Both sides are equal, confirming our solution is correct. Always perform this check to be certain that no mistakes were made along the way.

Equation Solving Steps

Understanding the sequence of steps used to solve equations is key. When tackling any equation, it's necessary to follow logical steps. Let's revisit the process using our example:

• Isolate the Square Root: Simplify the equation to just have the square root on one side.
• Square Both Sides: Once the square root is isolated, square both sides to remove the square root.
• Solve for the Variable: With the square root gone, solve the resulting simpler equation for the variable \(n\).
• Verify: Always verify the solution by substituting it back into the original equation.

These steps provide a structured approach to solving square root equations and can be applied to many similar problems.