Perform The Indicated Operations:$\[ \frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{x^2+7x-18} \\]

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying a complex algebraic expression involving fractions. We will break down the expression into manageable parts, apply the rules of algebra, and finally arrive at a simplified form.

The Given Expression

The given expression is:

2x−2+xx+9−x+20x2+7x−18\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{x^2+7x-18}

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of each fraction. The first fraction has a denominator of x−2x-2, which cannot be factored further. The second fraction has a denominator of x+9x+9, which also cannot be factored further. However, the third fraction has a denominator of x2+7x−18x^2+7x-18, which can be factored as (x−2)(x+9)(x-2)(x+9).

# **Factored Form of the Expression**

$\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{(x-2)(x+9)}$

Step 2: Find a Common Denominator

To add and subtract fractions, we need to have a common denominator. In this case, the common denominator is (x−2)(x+9)(x-2)(x+9). We can rewrite each fraction with this common denominator.

# **Rewritten Form of the Expression**

$\frac{2(x+9)}{(x-2)(x+9)} + \frac{x(x-2)}{(x-2)(x+9)} - \frac{x+20}{(x-2)(x+9)}$

Step 3: Combine the Fractions

Now that we have a common denominator, we can combine the fractions by adding and subtracting their numerators.

# **Combined Form of the Expression**

$\frac{2(x+9) + x(x-2) - (x+20)}{(x-2)(x+9)}$

Step 4: Simplify the Numerator

We can simplify the numerator by combining like terms.

# **Simplified Numerator**

$\frac{2x + 18 + x^2 - 2x - x - 20}{(x-2)(x+9)}$

Step 5: Final Simplification

We can simplify the numerator further by combining like terms.

# **Final Simplified Form**

$\frac{x^2 - 3}{(x-2)(x+9)}$

Conclusion

In this article, we simplified a complex algebraic expression involving fractions. We broke down the expression into manageable parts, applied the rules of algebra, and finally arrived at a simplified form. The final simplified form of the expression is x2−3(x−2)(x+9)\frac{x^2 - 3}{(x-2)(x+9)}. This expression cannot be simplified further, and it is the simplest form of the given expression.

Tips and Tricks

  • When simplifying algebraic expressions, it is essential to factor the denominators and find a common denominator.
  • When combining fractions, add and subtract their numerators.
  • When simplifying the numerator, combine like terms.
  • Finally, check if the expression can be simplified further.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. In engineering, algebraic expressions are used to design and optimize systems. In economics, algebraic expressions are used to model and analyze economic systems.

Common Mistakes

  • Not factoring the denominators.
  • Not finding a common denominator.
  • Not combining like terms in the numerator.
  • Not checking if the expression can be simplified further.

Practice Problems

  • Simplify the expression 3x+1+2x−2−1x2−3x−4\frac{3}{x+1} + \frac{2}{x-2} - \frac{1}{x^2-3x-4}.
  • Simplify the expression x+1x−2−x−1x+3+2x2−4x−5\frac{x+1}{x-2} - \frac{x-1}{x+3} + \frac{2}{x^2-4x-5}.

References

  • [1] "Algebra" by Michael Artin.
  • [2] "Calculus" by Michael Spivak.
  • [3] "Linear Algebra" by Jim Hefferon.

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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

A: The first step in simplifying an algebraic expression is to factor the denominators. This involves breaking down the denominator into its prime factors.

Q: How do I find a common denominator for fractions?

A: To find a common denominator for fractions, you need to identify the least common multiple (LCM) of the denominators. The LCM is the smallest multiple that all the denominators have in common.

Q: What is the difference between adding and subtracting fractions?

A: When adding fractions, you add the numerators and keep the common denominator. When subtracting fractions, you subtract the numerators and keep the common denominator.

Q: How do I simplify the numerator of a fraction?

A: To simplify the numerator of a fraction, you need to combine like terms. Like terms are terms that have the same variable raised to the same power.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check if the expression can be simplified further. This involves checking if there are any common factors in the numerator and denominator that can be canceled out.

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

A: Some common mistakes to avoid when simplifying algebraic expressions include not factoring the denominators, not finding a common denominator, not combining like terms in the numerator, and not checking if the expression can be simplified further.

Q: How do I know if an algebraic expression is in its simplest form?

A: An algebraic expression is in its simplest form when there are no common factors in the numerator and denominator that can be canceled out.

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

A: Simplifying algebraic expressions has numerous real-world applications, including physics, engineering, and economics.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through practice problems and exercises. You can also try simplifying expressions on your own and then checking your work with a calculator or a friend.

Q: What are some resources for learning more about simplifying algebraic expressions?

A: Some resources for learning more about simplifying algebraic expressions include textbooks, online tutorials, and practice problems. You can also try searching for videos and tutorials on YouTube and other online platforms.

Q: Can I simplify an algebraic expression that has a variable in the denominator?

A: Yes, you can simplify an algebraic expression that has a variable in the denominator. However, you need to be careful not to divide by zero.

Q: How do I simplify an algebraic expression that has a negative exponent?

A: To simplify an algebraic expression that has a negative exponent, you need to rewrite the expression with a positive exponent. This involves flipping the fraction and changing the sign of the exponent.

Q: Can I simplify an algebraic expression that has a fraction in the numerator?

A: Yes, you can simplify an algebraic expression that has a fraction in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a radical in the numerator?

A: To simplify an algebraic expression that has a radical in the numerator, you need to simplify the radical and then combine like terms.

Q: Can I simplify an algebraic expression that has a complex number in the numerator?

A: Yes, you can simplify an algebraic expression that has a complex number in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a trigonometric function in the numerator?

A: To simplify an algebraic expression that has a trigonometric function in the numerator, you need to use the trigonometric identities and then combine like terms.

Q: Can I simplify an algebraic expression that has a logarithmic function in the numerator?

A: Yes, you can simplify an algebraic expression that has a logarithmic function in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a combination of functions in the numerator?

A: To simplify an algebraic expression that has a combination of functions in the numerator, you need to use the function identities and then combine like terms.

Q: Can I simplify an algebraic expression that has a variable in the denominator and a fraction in the numerator?

A: Yes, you can simplify an algebraic expression that has a variable in the denominator and a fraction in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a negative exponent in the denominator?

A: To simplify an algebraic expression that has a negative exponent in the denominator, you need to rewrite the expression with a positive exponent. This involves flipping the fraction and changing the sign of the exponent.

Q: Can I simplify an algebraic expression that has a fraction in the denominator and a variable in the numerator?

A: Yes, you can simplify an algebraic expression that has a fraction in the denominator and a variable in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a radical in the denominator?

A: To simplify an algebraic expression that has a radical in the denominator, you need to simplify the radical and then combine like terms.

Q: Can I simplify an algebraic expression that has a complex number in the denominator?

A: Yes, you can simplify an algebraic expression that has a complex number in the denominator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a trigonometric function in the denominator?

A: To simplify an algebraic expression that has a trigonometric function in the denominator, you need to use the trigonometric identities and then combine like terms.

Q: Can I simplify an algebraic expression that has a logarithmic function in the denominator?

A: Yes, you can simplify an algebraic expression that has a logarithmic function in the denominator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a combination of functions in the denominator?

A: To simplify an algebraic expression that has a combination of functions in the denominator, you need to use the function identities and then combine like terms.

Q: Can I simplify an algebraic expression that has a variable in the denominator and a fraction in the numerator and a variable in the numerator?

A: Yes, you can simplify an algebraic expression that has a variable in the denominator and a fraction in the numerator and a variable in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a negative exponent in the denominator and a fraction in the numerator?

A: To simplify an algebraic expression that has a negative exponent in the denominator and a fraction in the numerator, you need to rewrite the expression with a positive exponent. This involves flipping the fraction and changing the sign of the exponent.

Q: Can I simplify an algebraic expression that has a fraction in the denominator and a variable in the numerator and a fraction in the numerator?

A: Yes, you can simplify an algebraic expression that has a fraction in the denominator and a variable in the numerator and a fraction in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a radical in the denominator and a fraction in the numerator?

A: To simplify an algebraic expression that has a radical in the denominator and a fraction in the numerator, you need to simplify the radical and then combine like terms.

Q: Can I simplify an algebraic expression that has a complex number in the denominator and a fraction in the numerator?

A: Yes, you can simplify an algebraic expression that has a complex number in the denominator and a fraction in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a trigonometric function in the denominator and a fraction in the numerator?

A: To simplify an algebraic expression that has a trigonometric function in the denominator and a fraction in the numerator, you need to use the trigonometric identities and then combine like terms.

Q: Can I simplify an algebraic expression that has a logarithmic function in the denominator and a fraction in the numerator?

A: Yes, you can simplify an algebraic expression that has a logarithmic function in the denominator and a fraction in the numerator. However, you need to be careful not to cancel out any common factors between the numerator and denominator.

Q: How do I simplify an algebraic expression that has a combination of functions in the denominator and a fraction in the numerator?

A: To simplify an algebraic expression that has a combination of functions