Problem 28 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 x+1}-2=0 $$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Isolate the Square Root

Start by isolating the square root term in the equation. To do this, add 2 to both sides of the equation: \( \sqrt{3x+1} = 2 \).

Step 2: Eliminate the Square Root

Next, eliminate the square root by squaring both sides of the equation. This gives: \( 3x + 1 = 4 \).

Step 3: Solve for x

To solve for \( x \), subtract 1 from both sides: \( 3x = 3 \). Then divide both sides by 3: \( x = 1 \).

Step 4: Check the Solution

Substitute \( x = 1 \) back into the original equation to verify the solution: \( \sqrt{3(1)+1} - 2 = 0 \). Simplifying gives: \( \sqrt{4} - 2 = 0 \), which is true since \( \sqrt{4} = 2 \).

Key Concepts

Square Roots

Square roots are an essential concept in algebra that can simplify complex problems and make them more manageable. Understanding square roots involve recognizing a number that, when multiplied by itself, gives the original number under the root. For example, \[ \sqrt{9} = 3 \]because 3 times 3 equals 9. Similarly, in the given problem, the square root plays a vital role in isolation and solution steps.

• To isolate a square root, one needs to consider moving other elements of the equation to the opposite side.
• When square roots are involved, they often form an equation that needs careful manipulation to solve.

Remember, the act of "squaring" both sides of an equation is the method to eliminate a square root. Let's consider our example: \[ \sqrt{3x+1} = 2 \]Squaring both sides leads us to:\[ 3x+1 = 4 \]This step is crucial as it simplifies the equation and brings clarity to the solution process.

Isolation of Terms

The technique of isolating terms forms the backbone of solving algebraic equations, such as when handling square roots. To isolate means to "unwind" the equation and shuffle its components so that you have the variable you are solving for on one side of the equation and other known quantities on the other side.

• Start by identifying the term you want to isolate; in many cases, this is under a square root, as seen in the given problem.
• Perform operations like addition or subtraction to both sides to shift unwanted terms.
• The goal is to have a clean expressions, like \( \sqrt{3x+1} = 2 \), where your term of interest is easily seen.

Remember, the process of isolation helps in making a complex expression simpler and in finding a concrete solution by dealing with one term at a time. This is why we added 2 to both sides in the example to isolate the root term first, leading us to the equation:\[3x = 3 \]which is far easier to solve.

Verification of Solutions

Verification of solutions is a critical final step in solving algebraic equations, ensuring the solution is indeed correct and satisfies the original problem. This step acts as a buffer against mistakes, serving to confirm that every manipulation of the terms led to a valid answer.

• Start by substituting the solution back into the original equation to check if both sides are equal.
• Perform the arithmetic to ensure consistency and correctness.

For instance, substitute \( x = 1 \) back into the original equation:\[\sqrt{3(1) + 1} - 2 = 0\]Calculate the inside of the root, giving \( \sqrt{4} \), which leads to:\[2 - 2 = 0\]This checks out perfectly, confirming our solution for \( x \) is accurate. Verification not only confirms accuracy but also provides confidence in your problem-solving skills, reinforcing the solution pathway of isolating terms and dealing with roots.