Problem 400 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{4 u+2}-6=0\)

Step-by-Step Solution

Answer

u = 8.5

Step 1: Isolate the Square Root

Add 6 to both sides of the equation to move the constant term to the right-hand side: \(\rightarrow \sqrt{4u + 2} - 6 + 6 = 0 + 6 \) Simplifies to: \( \rightarrow \sqrt{4u + 2} = 6 \)

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root: \(\rightarrow (\sqrt{4u + 2})^2 = 6^2 \) Simplifies to: \( \rightarrow 4u + 2 = 36 \)

Step 3: Solve for u

Subtract 2 from both sides to isolate the term with u: \( \rightarrow 4u + 2 - 2 = 36 - 2 \) Simplifies to: \( \rightarrow 4u = 34 \) Divide both sides by 4: \( \rightarrow \frac{4u}{4} = \frac{34}{4} \) Simplifies to: \( \rightarrow u = \frac{34}{4} = 8.5 \)

Step 4: Verify the Solution

Plug the value of u back into the original equation to verify the solution: \(\rightarrow \sqrt{4(8.5) + 2} - 6 = 0 \) Simplifies to: \(\rightarrow \sqrt{34 + 2} - 6 = 0 \) \(\rightarrow \sqrt{36} - 6 = 0 \) Since \( \sqrt{36} = 6 \), the left side becomes: \( 6 - 6 = 0 \), which is a true statement. Thus, u = 8.5 is the correct solution.

Key Concepts

Isolating the variable

When solving square root equations, the first crucial step is isolating the variable inside the square root. This means you need to move all other terms to the other side of the equation. In our example, we started with the equation \( \sqrt{4u + 2} - 6 = 0 \). To isolate the square root term, we added 6 to both sides, resulting in the simplified equation \( \sqrt{4u + 2} = 6 \). This makes it easier to handle the equation in later steps.

Squaring both sides

After isolating the square root term, the next step is to eliminate the square root by squaring both sides of the equation. In our example, we squared the equation \( \sqrt{4u + 2} = 6 \) to get \( \left( \sqrt{4u + 2} \right)^2 = 6^2 \), which simplifies to \( 4u + 2 = 36 \). Squaring both sides helps to transform the equation into a standard linear form, making it easier to solve for the variable.

Simplifying equations

Once we've squared both sides, we'll need to simplify the equation to solve for the variable. From \( 4u + 2 = 36 \), we first subtract 2 from each side, leading to \( 4u = 34 \). Then, by dividing both sides by 4, we isolate \( u \), resulting in \( u = 8.5 \). Simplifying equations is crucial as it hones down all necessary steps to isolate and solve for the variable.

Verifying solutions

Finally, always verify your solutions by plugging the variable back into the original equation. This ensures the solution is correct and consistent. For our example, substituting \( u = 8.5 \) back into the original equation \( \sqrt{4(8.5) + 2} - 6 = 0 \) gives \( \sqrt{36} - 6 = 0 \). Since \( \sqrt{36} = 6 \), we get \( 6 - 6 = 0 \), which is a true statement. Therefore, \( u = 8.5 \) is the verified solution.