Problem 20 - Algebra 1

Question

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

Step-by-Step Solution

Answer

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

Step 1: Isolate the square root term

This step is already taken care of in the existing equation, \(\sqrt{x} - 0 = 0\). Here, the square root term \(\sqrt{x}\) is already by itself on one side of the equation.

Step 2: Solve for the unknown variable x

Since the square root term \(\sqrt{x}\) equals 0, it implies that 'x' itself should be 0. This is because the square root of zero is zero.

Step 3: Check for extraneous solutions

In this step, substitute the obtained 'x' value (i.e., x=0) back into the original equation to see if it makes the equation true. On doing this, one gets: \(\sqrt{0} -0 = 0 - 0 = 0\), which hold true and thus, x = 0 is a valid solution.

Key Concepts

Extraneous Solutions in Solving Equations

When solving equations, especially those involving square roots, it's important to check for extraneous solutions. An extraneous solution is a root of the transformed equation that is not a solution to the original one. These can appear when both sides of an equation are squared while solving. This process can inadvertently introduce answers that do not actually satisfy the original equation.

• Extraneous solutions usually surface when the operation used in the transformation can introduce additional solutions. For instance, squaring both sides of an equation.
• Always re-substitute solutions back into the original equation to verify their validity.
• It ensures you're aware if the solution set may need to be adjusted, providing only correct answers.

Understanding extraneous solutions is pivotal for reliable problem-solving and avoiding unnecessary mistakes.

Understanding the Square Root in Equations

The square root of a number is a value that, when multiplied by itself, gives the original number. The operation of taking a square root is often seen in equations where the solution requires you to find a number whose square is equal to another number. It’s a fundamental operation in algebra dealing with powers and roots.

• In the equation \(\sqrt{x} - 0 = 0\), the square root operation is already isolated, which simplifies solving.
• Square roots only yield non-negative values in real numbers. This means if \sqrt{x}\ = 0, then x must equal zero.
• Learning to isolate and address the square root correctly makes handling more complex equations easier and more intuitive.

Square roots are a critical concept, offering a gateway to understanding more advanced mathematics, and recognizing their properties helps solve equations accurately.

Importance of Checking Solutions in Equations

After finding potential solutions, checking them is critical to confirm their accuracy. This is doubly important in equations with operations like squaring that can introduce extraneous results. Checking solutions involves substituting back into the original equation to confirm they satisfy it.

• This step eliminates any false solutions and verifies the true validity of each solution.
• In our exercise, verifying \(x = 0\) by substituting it back was straightforward, confirming its correctness.
• This process prevents errors and misunderstandings about which values truly solve the equation, solidifying your control over algebraic procedures.

Checking solutions rigorously ensures problem solutions are exact and reflects an in-depth understanding of mathematical concepts.