Problem 41 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x}-6=0 $$

Step-by-Step Solution

Answer

The solution to the equation \( \sqrt{x} - 6 = 0 \) is \( x = 36 \)

Step 1: Arrange the equation

The first step is to write the equation in the form you're familiar with:
\( \sqrt{x} = 6 \)

Step 2: Square both sides

Next, square both sides to remove the square root:
\( (\sqrt{x})^2 = 6^2 \)
This simplifies to:
\( x = 36 \)

Step 3: Check for extraneous solutions

Squaring both sides of an equation can potentially introduce extraneous solutions, so it's important to check. Substitute \(x = 36\) back into the original equation:
\( \sqrt{36} - 6 = 0 \)
This equation holds true, so \(x = 36\) is indeed a solution to the original equation and it's not extraneous.

Key Concepts

Extraneous Solutions

An extraneous solution is a solution that might seem correct at first glance but actually doesn't satisfy the original equation. Such solutions can emerge, especially when dealing with equations involving squares or square roots. This is because operations like squaring both sides of an equation can alter its fundamental properties.

In our example, after simplifying the equation \( \sqrt{x} = 6 \), we squared both sides to eliminate the square root, resulting in \( x = 36 \). However, this squaring process can sometimes introduce new solutions into the mix—solutions that weren't valid in the original context of the problem.

To ensure our solution isn't extraneous, we need to double-check by plugging it back into the original equation to see if it holds true. If it doesn't satisfy the original equation, then it's considered extraneous.

Checking Solutions

Checking solutions is a vital step in solving radical equations. This act ensures the accuracy and validity of any solutions you find, particularly since methods like squaring both sides can introduce extraneous solutions.

When you have found potential solutions, substitute them back into the starting equation. This checks they work within the original context. For our equation, the solution \( x = 36 \) gave us \( \sqrt{36} - 6 = 0 \). This satisfied the initial equation, meaning it was indeed a valid solution.

Consistently checking solutions helps prevent errors in more complex equations, ensuring that all solutions are truly valid. This step should become a natural part of your process when solving these types of equations.

Simplifying Equations

Simplifying an equation means breaking it down to make it easier to solve. In equations with square roots, like our example \( \sqrt{x} - 6 = 0 \), simplification often involves isolating the term under the square root.

Begin by rearranging to isolate the square root itself—\( \sqrt{x} = 6 \). Once isolated, perform operations (such as squaring) on both sides to remove the radical. This results in a simplified, straightforward equation (in this case, \( x = 36 \)) that often becomes much easier to solve.

Taking a complex or daunting equation and transforming it into something more manageable fits under this concept. Always ensure each simplification step is correct to maintain the equation's integrity.