Problem 18 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-10=0$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 100 \).

Step 1: Rearrange the Equation

First, add 10 to both sides of the equation \( \sqrt{x} - 10 = 0 \) to isolate the square root on one side. The equation becomes \( \sqrt{x} = 10 \).

Step 2: Solve for 'x'

Next, square both sides of the equation \( \sqrt{x} = 10 \) to solve for x. By squaring both sides, the equation becomes \( \left( \sqrt{x} \right)^2 = 10^2 \). It simplifies to \( x = 100 \).

Step 3: Check for Extraneous Solutions

To check whether \( x = 100 \) is a valid solution or an extraneous one, substitute \( x = 100 \) back into the original equation \( \sqrt{x} - 10 = 0 \). This results in \( \sqrt{100} - 10 = 0 \), which simplifies to \( 10 - 10 = 0 \). The left side equals to the right side, therefore, \( x = 100 \) is a valid solution and not an extraneous one.

Key Concepts

Extraneous Solutions

When solving square root equations, extraneous solutions can sometimes pop up. These solutions come from the process of squaring both sides of an equation to remove the square root. While this allows for solving the equation, it can also introduce solutions that don't actually satisfy the original problem. Hence, identifying extraneous solutions is crucial to ensure that the answers make sense in the context of the equation. In operations involving square roots, this step is important because squaring can introduce solutions that were not originally valid. In our problem, after solving, we found the solution as 100. However, it was essential to go back and verify if substituting 100 into the original equation would truly satisfy the equation. This checking step helps to identify if the derived solution is indeed correct or if it is extraneous.

Checking Solutions

Checking a solution in an equation is a necessary step in validating the findings. Once you have isolated a solution like we did with 100, you should always plug it back into the original equation to ensure correctness. This method is straightforward and confirms that no errors occurred during the solutions process. For our problem, substituting back, we used the original equation \( \sqrt{x} - 10 = 0 \). Substituting \( x = 100 \), we found that \( \sqrt{100} - 10 = 0 \) was true. Thus, confirming that \( x = 100 \) and it wasn't an extraneous solution. By verifying each solution, we ensure they work in the original context of the problem, which is a reliable tactic to confidently finalize solution results.

Step by Step Problem Solving

The step by step approach is a powerful technique to solve equations systematically, and it ensures understanding at each stage. Let's break down our approach:

• **Step 1:** Rearrange the equation to make it manageable. By adding 10 to both sides, we isolated the square root, which made it easier to manipulate.
• **Step 2:** Remove the square root by squaring both sides. This step opened the door to solving for \( x \) and revealed the solution \( x = 100 \).
• **Step 3:** Check for correctness. We substituted back to confirm our solution was true and not misleading or extraneous.

Focusing on the problem solving process can enhance your understanding of how equations work, especially with potentially complex operations like dealing with square roots. Each phase introduces a logical order and clarity, which is key to effective learning and application.