Problem 13 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ -5+\sqrt{x}=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 25\).

Step 1: Isolate the square root

Firstly, isolate the square root (\(\sqrt{x}\)) by adding 5 to both sides of the equation: \[ \sqrt{x} = 5 \]

Step 2: Remove the square root

To remove the square root, square both sides of the equation: \[ (\sqrt{x})^2 = 5^2 \implies x = 25 \]

Step 3: Check for extraneous solutions

Now, check the solution obtained in the original equation for any extraneous solutions. Substitute \(x=25\) into the original equation: \[ -5 + \sqrt{25} = -5 + 5 = 0 \] Since this is exactly what the original equation states, it verifies that \(x=25\) is indeed a solution and it is not extraneous.

Key Concepts

Understanding Extraneous Solutions

When solving square root equations, you might end up with solutions that do not actually satisfy the original equation. These are called extraneous solutions. Extraneous solutions often appear when both sides of an equation are squared during the solving process. This operation can sometimes introduce solutions that are not valid for the original equation. In the context of the original problem, once we square \(\sqrt{x} = 5\), we obtain \(x = 25\). To determine if this solution is extraneous, it must be checked against the original equation. This process ensures that the solution is valid and not merely an artifact of the equation-solving steps.

Isolate the Square Root

Before solving a square root equation, it is crucial to isolate the square root term. This means you want to get the square root by itself on one side of the equation. For instance, in the problem \(-5 + \sqrt{x} = 0\), we need to add 5 to both sides to isolate the square root: \(\sqrt{x} = 5\). By doing this, you have a simpler expression to work with, making the problem easier to solve. Once isolated, you can square both sides of the equation, which eliminates the square root. Keep in mind that any operations like squaring may introduce additional solutions, hence the importance of the next step to ensure accuracy.

Checking Solution for Validity

Once you have a potential solution from solving the equation, it's imperative to check its validity. Substitute this solution back into the original equation. This verification step ensures the solution satisfies the original problem and confirms it is not an extraneous solution.
For example, after isolating the square root and solving \(\sqrt{x} = 5\), we find \(x = 25\). By substituting back into the original equation: \(-5 + \sqrt{25} = 0\), and seeing that it holds true, we confirm \(x = 25\) is valid.
Always complete this step when solving square root equations to ensure no extraneous solutions are included in your final solution set.