Problem 20 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \sqrt{6 x+1}=5 $$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Isolate the Radical

The equation is \( \sqrt{6x + 1} = 5 \). The radical (square root) is already isolated on one side of the equation.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \( (\sqrt{6x + 1})^2 = 5^2 \). This simplifies to \( 6x + 1 = 25 \).

Step 3: Solve for x

Isolate \( x \) by first subtracting 1 from both sides: \( 6x + 1 - 1 = 25 - 1 \) gives \( 6x = 24 \). Next, divide both sides by 6: \( \frac{6x}{6} = \frac{24}{6} \) simplifies to \( x = 4 \).

Step 4: Verify the Solution

Substitute \( x = 4 \) back into the original equation to verify: \( \sqrt{6(4) + 1} = \sqrt{24 + 1} = \sqrt{25} = 5 \). Since both sides of the equation are equal when \( x = 4 \), the solution is correct.

Key Concepts

Solving Square Root Equations

Square root equations often appear intimidating, but they can be readily solved by following a systematic approach. These equations involve an expression inside a square root (or radical), which we aim to eliminate to solve the equation. Let's break it down even more simplistically:

• First, ensure the square root is alone on one side - this helps in simplifying the solution process.
• After isolating the radical, you'll need to square both sides to remove the square root.
• This process reduces the equation to a simpler algebraic form, usually involving a linear equation that can be solved for the variable of interest.

Think of square root equations as puzzles that require careful unraveling to reach the solution. By first clearing out the square root, we make the path toward solving for the variable clearer and devoid of confusion. Let's explore the first step in more detail by isolating the radical.

Isolating the Radical

The initial step in solving a square root equation is all about setting the stage for simplification. When we talk about isolating the radical, we mean ensuring that the square root is the only term on its side of the equation, without any coefficients multiplying or numbers adding to it.

In our example, you can see that \( \sqrt{6x + 1} = 5 \) already has the square root isolated. This is ideal because it allows us to immediately proceed to the next step without any additional work to consolidate the terms.

• Look for equations where you might need to move terms around or divide by a coefficient to get the radical alone.
• Make sure nothing else modifies the square root term; it should be as standalone as possible.

Once isolated, the equation is ripe for squaring both sides, which will eliminate the square root and simplify the path to finding the solution. So, the tedious work upfront pays off with ease in subsequent steps.

Verifying Solutions in Algebra

Verifying solutions is a crucial part of solving equations, particularly when dealing with square roots. Sometimes, the process of squaring both sides might introduce extraneous solutions - solutions that, while mathematically valid, do not actually satisfy the original equation.

To check your work, plug the value of the variable back into the original equation. Let's see how it's done using our example:

• Take \( x = 4 \) and substitute it back into \( \sqrt{6(4) + 1} = 5 \).
• This simplifies to \( \sqrt{24 + 1} = \sqrt{25} = 5 \).
• Because both sides of the equation match, you can be confident that \( x = 4 \) is indeed a valid solution.

Verification ensures reliability in your solution, confirming that you have not only arrived at a mathematically sound answer but that it also adheres to the original equation's constraints. This step protects against errors made during the algebraic manipulation process and instills confidence in the solution obtained.