A Student Solves The Following Equation And Determines That The Solution Is \[$-2\$\]. Is The Student Correct? Explain.$\[ \frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2} \\]

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A Critical Analysis of the Student's Solution

In mathematics, solving equations is a fundamental concept that requires a deep understanding of algebraic manipulations and logical reasoning. A student who claims to have solved the equation 3a+2−6aa2−4=1a−2\frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2} and determined that the solution is −2-2 may seem confident, but is the student correct? In this article, we will delve into the equation, analyze the student's solution, and provide a critical evaluation of their work.

The given equation is 3a+2−6aa2−4=1a−2\frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2}. To begin with, we need to simplify the equation by finding a common denominator. The common denominator for the left-hand side of the equation is (a+2)(a−2)(a+2)(a-2), which can be factored as (a2−4)(a^2-4). Therefore, the equation becomes:

3(a−2)(a+2)(a−2)−6a(a+2)(a2−4)(a+2)=1a−2\frac{3(a-2)}{(a+2)(a-2)} - \frac{6a(a+2)}{(a^2-4)(a+2)} = \frac{1}{a-2}

Now that we have a common denominator, we can simplify the equation by combining the fractions on the left-hand side:

3(a−2)−6a(a+2)(a2−4)=1a−2\frac{3(a-2) - 6a(a+2)}{(a^2-4)} = \frac{1}{a-2}

Expanding the numerator, we get:

3a−6−6a2−12a(a2−4)=1a−2\frac{3a - 6 - 6a^2 - 12a}{(a^2-4)} = \frac{1}{a-2}

Combining like terms, we have:

−6a2−9a−6(a2−4)=1a−2\frac{-6a^2 - 9a - 6}{(a^2-4)} = \frac{1}{a-2}

The student claims that the solution to the equation is −2-2. To verify this, we need to substitute −2-2 into the original equation and check if it satisfies the equation. Substituting a=−2a = -2 into the equation, we get:

3−2+2−6(−2)(−2)2−4=1−2−2\frac{3}{-2+2} - \frac{6(-2)}{(-2)^2-4} = \frac{1}{-2-2}

Simplifying the equation, we get:

30−−120=1−4\frac{3}{0} - \frac{-12}{0} = \frac{1}{-4}

This is an undefined expression, as division by zero is not allowed in mathematics. Therefore, the student's solution of −2-2 is incorrect.

In conclusion, the student's solution of −2-2 is incorrect because it leads to an undefined expression. To solve the equation correctly, we need to simplify the equation, find a common denominator, and then substitute the solution into the original equation to verify its validity. The correct solution to the equation is not −2-2, but rather a value that satisfies the equation and does not lead to an undefined expression.

When solving equations, it is essential to avoid common mistakes that can lead to incorrect solutions. Some of these mistakes include:

  • Not finding a common denominator: Failing to find a common denominator can lead to incorrect simplifications and solutions.
  • Not checking for undefined expressions: Failing to check for undefined expressions can lead to incorrect solutions and conclusions.
  • Not verifying the solution: Failing to verify the solution by substituting it into the original equation can lead to incorrect conclusions.

To solve equations correctly, it is essential to follow best practices that ensure accuracy and validity. Some of these best practices include:

  • Simplifying the equation: Simplifying the equation by finding a common denominator and combining fractions can make it easier to solve.
  • Checking for undefined expressions: Checking for undefined expressions can help identify potential errors and ensure that the solution is valid.
  • Verifying the solution: Verifying the solution by substituting it into the original equation can ensure that the solution is correct and valid.

By following these best practices and avoiding common mistakes, you can ensure that your solutions are accurate and valid, and that you are well on your way to becoming a proficient mathematician.
A Student's Guide to Solving Equations: Q&A

In our previous article, we analyzed the equation 3a+2−6aa2−4=1a−2\frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2} and determined that the student's solution of −2-2 was incorrect. In this article, we will provide a Q&A guide to help students understand the concepts and techniques involved in solving equations.

A: The first step in solving an equation is to simplify the equation by finding a common denominator. This involves identifying the denominators of the fractions in the equation and finding a common factor that can be used to combine the fractions.

A: To find a common denominator, you need to identify the denominators of the fractions in the equation and find a common factor that can be used to combine the fractions. For example, in the equation 3a+2−6aa2−4=1a−2\frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2}, the common denominator is (a+2)(a−2)(a+2)(a-2), which can be factored as (a2−4)(a^2-4).

A: After finding a common denominator, the next step is to combine the fractions by multiplying the numerators and denominators of each fraction by the necessary factors to obtain a common denominator.

A: To combine fractions, you need to multiply the numerators and denominators of each fraction by the necessary factors to obtain a common denominator. For example, in the equation 3a+2−6aa2−4=1a−2\frac{3}{a+2} - \frac{6a}{a^2-4} = \frac{1}{a-2}, the combined fraction is 3(a−2)−6a(a+2)(a2−4)\frac{3(a-2) - 6a(a+2)}{(a^2-4)}.

A: The final step in solving an equation is to verify the solution by substituting it into the original equation. This ensures that the solution is correct and valid.

A: Verifying the solution is important because it ensures that the solution is correct and valid. If the solution is not verified, it may lead to incorrect conclusions and mistakes.

A: Some common mistakes to avoid when solving equations include:

  • Not finding a common denominator: Failing to find a common denominator can lead to incorrect simplifications and solutions.
  • Not checking for undefined expressions: Failing to check for undefined expressions can lead to incorrect solutions and conclusions.
  • Not verifying the solution: Failing to verify the solution by substituting it into the original equation can lead to incorrect conclusions.

A: To improve your skills in solving equations, you can:

  • Practice regularly: Practice solving equations regularly to develop your skills and build your confidence.
  • Seek help when needed: Seek help from teachers, tutors, or classmates when you need it.
  • Review and practice: Review and practice solving equations to reinforce your understanding and build your skills.

By following these tips and avoiding common mistakes, you can improve your skills in solving equations and become a proficient mathematician.