Simplify The Expression:$\[ \frac{2}{x^2-16} - \frac{4}{x^2+x-12} \\]

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Introduction

Algebraic expressions can be complex and daunting, especially when dealing with fractions and variables. In this article, we will focus on simplifying a given expression involving fractions and variables. We will break down the problem into manageable steps, making it easier to understand and solve.

The Given Expression

The given expression is:

2x2βˆ’16βˆ’4x2+xβˆ’12\frac{2}{x^2-16} - \frac{4}{x^2+x-12}

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators. The first denominator, x2βˆ’16x^2-16, can be factored as:

x2βˆ’16=(x+4)(xβˆ’4)x^2-16 = (x+4)(x-4)

The second denominator, x2+xβˆ’12x^2+x-12, can be factored as:

x2+xβˆ’12=(x+4)(xβˆ’3)x^2+x-12 = (x+4)(x-3)

Step 2: Rewrite the Expression with Factored Denominators

Now that we have factored the denominators, we can rewrite the expression as:

2(x+4)(xβˆ’4)βˆ’4(x+4)(xβˆ’3)\frac{2}{(x+4)(x-4)} - \frac{4}{(x+4)(x-3)}

Step 3: Find a Common Denominator

To combine the fractions, we need to find a common denominator. The least common multiple (LCM) of (x+4)(xβˆ’4)(x+4)(x-4) and (x+4)(xβˆ’3)(x+4)(x-3) is (x+4)(xβˆ’4)(xβˆ’3)(x+4)(x-4)(x-3).

Step 4: Rewrite the Expression with a Common Denominator

Now that we have found the common denominator, we can rewrite the expression as:

2(xβˆ’3)(x+4)(xβˆ’4)(xβˆ’3)βˆ’4(xβˆ’4)(x+4)(xβˆ’4)(xβˆ’3)\frac{2(x-3)}{(x+4)(x-4)(x-3)} - \frac{4(x-4)}{(x+4)(x-4)(x-3)}

Step 5: Combine the Fractions

Now that we have a common denominator, we can combine the fractions by adding or subtracting the numerators:

2(xβˆ’3)βˆ’4(xβˆ’4)(x+4)(xβˆ’4)(xβˆ’3)\frac{2(x-3) - 4(x-4)}{(x+4)(x-4)(x-3)}

Step 6: Simplify the Numerator

We can simplify the numerator by distributing the negative sign and combining like terms:

2xβˆ’6βˆ’4x+16(x+4)(xβˆ’4)(xβˆ’3)\frac{2x-6-4x+16}{(x+4)(x-4)(x-3)}

βˆ’2x+10(x+4)(xβˆ’4)(xβˆ’3)\frac{-2x+10}{(x+4)(x-4)(x-3)}

Step 7: Factor the Numerator

We can factor the numerator as:

βˆ’2(xβˆ’5)(x+4)(xβˆ’4)(xβˆ’3)\frac{-2(x-5)}{(x+4)(x-4)(x-3)}

Conclusion

In this article, we simplified a complex algebraic expression involving fractions and variables. We broke down the problem into manageable steps, making it easier to understand and solve. By factoring the denominators, finding a common denominator, combining the fractions, simplifying the numerator, and factoring the numerator, we were able to simplify the expression.

Final Answer

The simplified expression is:

βˆ’2(xβˆ’5)(x+4)(xβˆ’4)(xβˆ’3)\frac{-2(x-5)}{(x+4)(x-4)(x-3)}

Tips and Tricks

  • When dealing with complex algebraic expressions, break down the problem into manageable steps.
  • Factor the denominators to simplify the expression.
  • Find a common denominator to combine the fractions.
  • Simplify the numerator by distributing the negative sign and combining like terms.
  • Factor the numerator to simplify the expression further.

Common Mistakes

  • Failing to factor the denominators.
  • Not finding a common denominator.
  • Not simplifying the numerator.
  • Not factoring the numerator.

Real-World Applications

Simplifying complex algebraic expressions has many real-world applications, such as:

  • Physics: Simplifying expressions to describe the motion of objects.
  • Engineering: Simplifying expressions to design and optimize systems.
  • Economics: Simplifying expressions to model and analyze economic systems.

Conclusion

Introduction

In our previous article, we simplified a complex algebraic expression involving fractions and variables. We broke down the problem into manageable steps, making it easier to understand and solve. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex algebraic expressions.

Q&A

Q: What is the first step in simplifying a complex algebraic expression?

A: The first step in simplifying a complex algebraic expression is to factor the denominators. This will help to simplify the expression and make it easier to work with.

Q: How do I factor the denominators?

A: To factor the denominators, you need to find the greatest common factor (GCF) of the terms in the denominator. You can then use the GCF to factor the denominator into simpler terms.

Q: What is the least common multiple (LCM) and how do I find it?

A: The LCM is the smallest multiple that two or more numbers have in common. To find the LCM, you can list the multiples of each number and find the smallest multiple that they have in common.

Q: How do I combine fractions with different denominators?

A: To combine fractions with different denominators, you need to find a common denominator. You can then rewrite each fraction with the common denominator and add or subtract the numerators.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top part of a fraction, and the denominator is the bottom part. The numerator is the number being divided, and the denominator is the number by which we are dividing.

Q: How do I simplify a complex algebraic expression with multiple variables?

A: To simplify a complex algebraic expression with multiple variables, you need to follow the same steps as before. Factor the denominators, find a common denominator, combine the fractions, and simplify the numerator.

Q: What are some common mistakes to avoid when simplifying complex algebraic expressions?

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

  • Failing to factor the denominators
  • Not finding a common denominator
  • Not simplifying the numerator
  • Not factoring the numerator

Q: How do I check my work when simplifying a complex algebraic expression?

A: To check your work when simplifying a complex algebraic expression, you can plug in a value for the variable and see if the expression simplifies to the expected value.

Q: What are some real-world applications of simplifying complex algebraic expressions?

A: Some real-world applications of simplifying complex algebraic expressions include:

  • Physics: Simplifying expressions to describe the motion of objects
  • Engineering: Simplifying expressions to design and optimize systems
  • Economics: Simplifying expressions to model and analyze economic systems

Conclusion

Simplifying complex algebraic expressions is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify even the most complex expressions. Remember to check your work and consider real-world applications to ensure that you are using your skills effectively.

Additional Resources

  • Khan Academy: Simplifying Algebraic Expressions
  • Mathway: Simplifying Algebraic Expressions
  • Wolfram Alpha: Simplifying Algebraic Expressions

Practice Problems

  • Simplify the expression: 2x2βˆ’16βˆ’4x2+xβˆ’12\frac{2}{x^2-16} - \frac{4}{x^2+x-12}
  • Simplify the expression: 3x2+5x+6βˆ’2x2+7x+12\frac{3}{x^2+5x+6} - \frac{2}{x^2+7x+12}
  • Simplify the expression: 4x2βˆ’9βˆ’3x2βˆ’4\frac{4}{x^2-9} - \frac{3}{x^2-4}

Answer Key

  • Simplify the expression: βˆ’2(xβˆ’5)(x+4)(xβˆ’4)(xβˆ’3)\frac{-2(x-5)}{(x+4)(x-4)(x-3)}
  • Simplify the expression: βˆ’5(x+3)(x+2)\frac{-5}{(x+3)(x+2)}
  • Simplify the expression: 7(xβˆ’3)(x+3)\frac{7}{(x-3)(x+3)}