Problem 46 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check for extraneous solutions. $$ \sqrt{x}+11=1 $$

Step-by-Step Solution

Answer

The equation \(\sqrt{x} + 11 = 1\) has no solutions because its only potential solution \(x = 100\) is extraneous.

Step 1: Isolate the Square Root

The first step is to isolate the square root. This can be accomplished by subtracting 11 from both sides of the equation. The resulting equation is: \(\sqrt{x}+11-11 = 1-11\) Simplifying this equation gives \(\sqrt{x} = -10\)

Step 2: Square Both Sides

Next, square both sides of the equation to get rid of the square root. This gives: \((\sqrt{x})^2 = (-10)^2\) which simplifies to \(x = 100\)

Step 3: Check for Extraneous Solutions

Finally, substitute \(x = 100\) into the original equation to check for any extraneous solutions. Doing this gives: \(\sqrt{100} + 11 = 1\) Simplify further to get \(10 + 11 = 1\), which is not true. So, \(x = 100\) is an extraneous solution.

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of mathematics, involving finding the value of the unknowns that make the equation true. In the context of the provided exercise, we needed to solve for the variable \(x\) in the equation \(\sqrt{x} + 11 = 1\). To do this, we perform operations that simplify the equation and isolate \(x\).

The process of solving any algebraic equation often involves the following general steps:

• Identify the variable to solve for and rearrange the equation to isolate it on one side if needed.
• Apply inverse operations to eliminate terms connected to the variable.
• Simplify the equation through arithmetic or algebraic operations.

In our example, isolating \(x\) involves dealing with the square root term and the constant (11). Subtract 11 from both sides to begin unraveling the equation. This step is crucial, as it sets the stage for more advanced algebraic manipulations like squaring, needed when dealing with square roots.

Square Root Isolation

Isolating the square root is an important step when dealing with equations that include terms like \(\sqrt{x}\). The isolation process helps simplify the equation and make the next steps more straightforward.

When you start with \(\sqrt{x} + 11 = 1\), the goal is to have the square root \(\sqrt{x}\) by itself on one side of the equation. This means you need to subtract 11 from both sides of the equation:

• Start with \(\sqrt{x} + 11 = 1\).
• Subtract 11 from both sides, yielding \(\sqrt{x} = 1 - 11\).
• Simplify to \(\sqrt{x} = -10\).

After isolating the square root, the next logical step is to square both sides of the equation to remove the square root entirely. However, in this case, immediately after isolation, it's clear there's an issue: the square root of a number cannot be negative. When you see \(\sqrt{x} = -10\), you should recognize that there might be extraneous solutions, meaning possible results that don't actually satisfy the original equation.

Checking Solutions

Checking solutions is a vital step to ensure the accuracy of your results after solving an equation. In equations involving square roots, this is particularly important because the operations performed can sometimes produce extraneous solutions—solutions that are mathematically generated during the solving process but do not satisfy the original equation.

Here's how you check for these solutions:

• After solving for \(x\) and possibly finding \(x = 100\), substitute back into the original equation.
• In this case, substitute \(x = 100\) into \(\sqrt{x} + 11 = 1\), giving \(\sqrt{100} + 11\).
• Simplify to find \(10 + 11 = 21\).
• Since 21 does not equal 1, \(x = 100\) does not satisfy the original equation.

This discrepancy shows that \(x = 100\) is an extraneous solution. Therefore, always perform this check to validate your solutions, especially in cases dealing with operations that can introduce unwanted results, such as squaring both sides of an equation. This careful check avoids errors and ensures the solutions you find truly meet the problem's requirements.