Problem 22 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{4 x+1}+5=10$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 6 \)

Step 1: Isolate the square root term

We start by isolating the square root term \( \sqrt{4 x+1} \) on one side of the equation by subtracting 5 from both sides of the equation: \( \sqrt{4 x+1} = 10 - 5 \). Simplifying, we get \( \sqrt{4 x+1} = 5 \).

Step 2: Eliminate the square root

Next, we eliminate the square root by squaring both sides of the equation: \( (\sqrt{4 x+1})^2 = 5^2 \). Simplifying, this yields \( 4x + 1 = 25 \).

Step 3: Solve for x

We solve the equation for x by subtracting 1 from both sides, getting \( 4x = 24 \), and then dividing each side by 4, yielding \( x = 6 \).

Step 4: Check for extraneous solutions

We check the obtained solution by substituting \( x = 6 \) back into the original equation. This gives us \( \sqrt{4*6+1} + 5 \). Simplifying, we get \( \sqrt{25} + 5 = 10 \), and since 10 = 10, we find that x = 6 is a valid solution of the original equation. The equation indeed has no extraneous solutions.

Key Concepts

Understanding Square Roots

To solve equations involving square roots, it's crucial to understand what a square root signifies. A square root of a number is a value that, when multiplied by itself, returns the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. When solving equations like \(\sqrt{4x+1}+5=10\), the first step is often to isolate the square root term. This makes it easier to remove the square root by squaring both sides, simplifying the equation to something more familiar.

Here’s a helpful sequence for dealing with square roots:

• Isolate the square root on one side of the equation.
• Eliminate the square root by squaring both sides.
• Solve the resulting equation.

By following these steps, you can effectively manage square roots and solve related equations efficiently. Remember, isolating the square root prepares it for squaring, which simplifies the task of solving the rest of the equation.

Extraneous Solutions Exploration

Extraneous solutions often occur when solving equations involving square roots. These are solutions that emerge during the algebraic process but aren't true solutions to the original equation.

When you square both sides of an equation to eliminate square roots, additional solutions might be introduced. That's why it's crucial to check every solution by substituting it back into the original equation to ensure it holds true.

For instance, after finding \(x = 6\), substitute it back into the equation \(\sqrt{4x+1}+5=10\). Calculate to verify if both sides are equal. If so, the solution is valid; if not, it's extraneous. Being vigilant about checking for extraneous solutions is essential for accuracy in solving equations with square roots.

Isolation of Variables

The concept of isolating variables is fundamental in algebra, especially when dealing with equations containing multiple terms and operations. To isolate a term means to rearrange the equation in such a way that you have the unknown variable on one side and everything else on the other. This process can include operations such as addition, subtraction, multiplication, and division.

In our example \(\sqrt{4x+1}+5=10\), isolating \(\sqrt{4x+1}\) involves subtracting 5 from both sides, leaving \(\sqrt{4x+1} = 5\). With the square root isolated, it becomes easier to solve for \(x\) by squaring both sides.

To isolate a variable:

• Perform inverse operations to move other terms to the opposite side of the equation.
• Maintain balance by performing the same operation on both sides.
• Continue until the variable is alone on one side of the equation.

Mastering the isolation of variables is pivotal in solving equations efficiently and forms the backbone of algebraic problem-solving.