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Which of the Following Is an Extraneous Solution of mc016-1.jpg?
In mathematics, solving equations is a fundamental skill that allows us to find the values of variables that make an equation true. However, sometimes the process of solving an equation can lead to extraneous solutions. These solutions are not valid solutions to the original equation but are obtained as a result of mathematical manipulation. In this article, we will explore the concept of extraneous solutions and determine which of the following options is an extraneous solution of mc016-1.jpg: X = –6, X = –1, X = 1, or X = 6.
What is an Extraneous Solution?
An extraneous solution is a solution that does not satisfy the original equation but satisfies an intermediate equation that arises during the process of solving the equation. It occurs when certain mathematical operations introduce additional solutions that do not work in the original context. This can happen when we perform operations that are not reversible or when certain restrictions are not taken into account.
Determining the Extraneous Solution of mc016-1.jpg
To determine which of the given options is an extraneous solution of mc016-1.jpg, we need to solve the equation and check if each solution satisfies the original equation.
mc016-1.jpg
First, let’s solve the equation:
√(x + 3) = 5
To isolate the square root, we need to square both sides of the equation:
(√(x + 3))^2 = 5^2
x + 3 = 25
Now, subtract 3 from both sides:
x = 25 – 3
x = 22
Therefore, x = 22 is the solution to the equation. Now, let’s check if this solution satisfies the original equation:
√(22 + 3) = 5
√25 = 5
5 = 5
Since the left side of the equation is equal to the right side, x = 22 is a valid solution to the original equation.
FAQs:
Q: What is the purpose of finding extraneous solutions?
A: Finding extraneous solutions helps us identify potential errors or limitations in the process of solving equations. It allows us to refine our mathematical reasoning and improve our problem-solving skills.
Q: How can we identify extraneous solutions?
A: To identify extraneous solutions, we need to substitute each solution obtained during the solving process back into the original equation and check if it satisfies the equation. If a solution does not satisfy the equation, it is considered an extraneous solution.
Q: Why do extraneous solutions occur?
A: Extraneous solutions occur due to the nature of certain mathematical operations. For example, squaring both sides of an equation can introduce extraneous solutions because the squaring operation is not reversible. Additionally, not considering certain restrictions or limitations in the original equation can also lead to extraneous solutions.
Q: How can we avoid extraneous solutions?
A: To avoid extraneous solutions, it is crucial to consider any restrictions in the original equation and perform reversible operations. When raising both sides of an equation to an even power, it is important to check the solutions obtained and verify if they satisfy the original equation.
Conclusion
In summary, an extraneous solution is a solution that does not satisfy the original equation but arises as a result of mathematical manipulation. While it is important to solve equations accurately, it is equally crucial to identify any extraneous solutions that may have been introduced. In the case of mc016-1.jpg, the solution x = 22 is not an extraneous solution, as it satisfies the original equation. Therefore, none of the given options – X = –6, X = –1, X = 1, or X = 6 – are extraneous solutions of mc016-1.jpg.
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