Problem 69 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Explain what is wrong with the following steps for solving a radical equation. $$\begin{aligned}\sqrt{x+1}-2 &=0 \\\\(x+1)+4 &=0 \\\x &=-5\end{aligned}$$

Step-by-Step Solution

Answer

The main error lies in the second step where they added 4 to \(x+1\) instead of squaring both sides of the equation. This makes the final solution incorrect since it is based on an incorrect step.

Step 1: Check the Initial Equation

The initial equation is \(\sqrt{x+1}-2 =0 \). The goal here is to isolate the square root on one side of the equation and solve for the variable x. This step is correct and doesn't show any error.

Step 2: Assess the Second Step

The next step provided is \((x+1)+4=0\). The error lies here, as the operation performed on the initial equation isn't correct. Instead of adding the square of the number subtracted from the square root to both sides of the equation (which would be the appropriate step while solving a radical equation like this), they simply added 4 to \(x+1\). The correct step would be square both sides of the equation.

Step 3: Verify the Last Step

The last step of the solution is x=-5, which is derived based on the error in the second step. As the second step is incorrect, this solution doesn't follow and does not correctly solve for x.

Key Concepts

Common Errors in Algebra

Algebra often trips up students because of subtle errors that can affect the outcome of a problem. One common error occurs when dealing with radical equations—equations involving roots such as square roots. A typical mistake is transforming the equation incorrectly during the solution process. For instance, in the provided exercise, the error happens during the transition from \( \sqrt{x+1}-2 = 0 \) to \((x+1)+4 = 0\). The error originates from trying to manipulate the equation without correctly addressing the square root.

When solving radical equations, it is crucial first to isolate the radical term, and then square both sides of the equation to eliminate the square root. By avoiding improper operations, such as arbitrary additions or subtractions, you can follow a logical pathway to the correct solution.

To prevent these common mistakes:

• Ensure you understand the operations needed to isolate the variable.
• Double-check your steps before moving on to the next part of the solution.
• Remember that failing to correctly transform the equations often leads to incorrect solutions.

Steps for Solving Equations

Solving equations, especially those involving radicals, requires a precise approach. Let's break down the effective steps to solve the original equation \(\sqrt{x+1}-2 = 0\):

1. **Isolate the Radical:**
Move the constant term to the other side of the equation. For our equation, add 2 to both sides, resulting in \(\sqrt{x+1} = 2\).
2. **Eliminate the Radical:**
Square both sides of the equation to remove the square root. This gives us \(x+1 = 4\).
3. **Solve for the Variable:**
Subtract 1 from both sides to solve for \(x\). This leaves \(x = 3\).

By following these steps, you ensure that you carefully handle the operation necessary to solve the equation correctly. This method avoids mistakes and helps you arrive at the correct solution efficiently.

Mathematical Verification

Once you have solved an equation, especially one involving radicals, it's crucial to verify your solution. Verification checks if the solution satisfies the original equation. In case of our original problem, after solving for \(x = 3\), we need to plug it back into the original equation to confirm its accuracy.

Substituting \(x = 3\) back into the equation \(\sqrt{x+1} - 2 = 0\):

• Calculate \(\sqrt{3+1} = \sqrt{4} = 2\).
• Check whether \(2 - 2 = 0\), which confirms the solution is correct.

If the equation holds true, then the solution is verified as accurate. This step is essential since radical equations can sometimes give misleading solutions. Real and extraneous solutions sometimes arise, so never skip verification. If it doesn’t check out, it's worth revisiting your steps to find where the error occurred.