Problem 126 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After squaring both sides of a radical equation, the only solution that I obtained was extrancous, so \(\varnothing\) must be the solution set of the original equation.

Step-by-Step Solution

Answer

Without more context, this statement can't be definitively evaluated. While an extraneous solution was identified correctly, it doesn't automatically mean that the original radical equation has no valid solutions, unless all potential solutions were explored and determined to be extraneous. As such, while not entirely baseless, this statement isn't fully logical or necessarily correct.

Step 1: Understand the statement

First, let's analyze the statement given. It refers to the process of squaring both sides of a radical equation, which is a common step in solving such equations. After doing so, the speaker has obtained what they identified as an extraneous solution, which meant that the solution they obtained isn't valid for the original equation. They've then concluded that the solution set of the entire original equation is an empty set or \(\varnothing\), implying that there are no solutions.

Step 2: Analyze the logic behind the statement

An extraneous solution is a 'false' solution that doesn't satisfy the original equation. These often arise due to the squaring process, as squaring removes the 'sign' information (whether a number was originally positive or negative). However, the presence of an extraneous solution doesn't automatically mean that no valid solutions exist, unless it was shown that the extracted solution was the only possible one. The speaker didn't specify whether they checked for other solutions or just found an extraneous one and assumed there were no others.

Step 3: Come to a conclusion

Without more context, we can't definitively say whether the speaker's conclusion about the original solution set is valid or not. While their process to check for an extraneous solution was correct, they would need to ensure that no other potential solutions exist before concluding that the original equation has no solutions. The presence of an extraneous solution doesn't automatically negate the possibility of other, valid solutions. So, the statement doesn't necessarily make sense as it stands, as the conclusion could still be premature and not entirely logical.

Key Concepts

Extraneous Solutions

When solving radical equations, extraneous solutions can pop up unexpectedly. These are solutions that emerge from the process of solving the equation but do not actually satisfy the original equation. This happens a lot when you square both sides of an equation, as squaring can introduce solutions that aren't truly viable.
One reason extraneous solutions appear is because squaring both sides of an equation results in losing information about the original sign of the numbers. For example, if you have a solution that was supposed to be positive but squaring makes it show as a valid negative solution, ruining its validity in the original equation.

• They often do not satisfy the original equation.
• They appear frequently when dealing with squared equations.
• It is crucial to check each solution in the original equation to verify its validity.

Whenever you solve a radical equation and find a solution, take a minute to plug it back into the initial equation. If it doesn't work, you've hit an extraneous solution.

Solution Set

The solution set of an equation is essentially the set of all numbers that satisfy the original equation. When solving radical equations, it is key to determine the complete and correct solution set to ensure no solutions are overlooked or falsely included. In the scenario where the only solution found is extraneous, the solution set might be empty, represented by \(\varnothing\).
If you conclude that the solution set is empty after finding an extraneous solution, there are a few things you should double-check:

• Verify that you correctly simplified and squared the equation.
• Ensure that you haven’t overlooked any steps that could lead to additional solutions.
• Consider any potential errors in your calculations.

Only when you’re certain that there are no valid solutions is it appropriate to declare the solution set as empty.

Squaring Equations

Squaring both sides of a radical equation is a common method to eliminate the radical, allowing easier solution of the equation. However, this process can introduce extraneous solutions, so it should be done carefully.
When squaring an equation:

• Ensure that each side is correctly squared. This means squaring every term on the left and right sides.
• Keep in mind that squaring can sometimes result in more than one solution. Always test each one in the original equation.
• Be vigilant about potential solution loss or addition due to squaring.

By proceeding with care, squaring equations can be a powerful tool, but remember to always verify your solutions in the initial equation to avoid errors.