Simplify The Expression: X X − 6 − 3 X + 6 \frac{x}{x-6} - \frac{3}{x+6} X − 6 X − X + 6 3

Mar 13, 2025 by ADMIN 95 views

$Simplify the Expression: $\frac{x}{x-6} - \frac{3}{x+6}$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we will focus on simplifying the given expression: $\frac{x}{x-6} - \frac{3}{x+6}$ . We will use various techniques, including finding a common denominator, combining fractions, and factoring, to simplify the expression.

Understanding the Expression

The given expression is a difference of two fractions: $\frac{x}{x-6} - \frac{3}{x+6}$ . To simplify this expression, we need to find a common denominator, which is the least common multiple (LCM) of the denominators. In this case, the LCM of $(x-6)$ and $(x+6)$ is $(x-6)(x+6)$ .

Finding a Common Denominator

To find a common denominator, we need to multiply the numerator and denominator of each fraction by the necessary factors. For the first fraction, we need to multiply the numerator and denominator by $(x+6)$ , while for the second fraction, we need to multiply the numerator and denominator by $(x-6)$ .

\frac{x}{x-6} - \frac{3}{x+6} = \frac{x(x+6)}{(x-6)(x+6)} - \frac{3(x-6)}{(x-6)(x+6)}

Combining Fractions

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

\frac{x(x+6)}{(x-6)(x+6)} - \frac{3(x-6)}{(x-6)(x+6)} = \frac{x(x+6) - 3(x-6)}{(x-6)(x+6)}

Simplifying the Numerator

To simplify the numerator, we need to expand and combine like terms.

x(x+6) - 3(x-6) = x^2 + 6x - 3x + 18

Factoring the Numerator

Now that we have simplified the numerator, we can factor it to see if there are any common factors that can be canceled out.

x^2 + 6x - 3x + 18 = x^2 + 3x + 18

Canceling Out Common Factors

Now that we have factored the numerator, we can cancel out any common factors between the numerator and denominator.

\frac{x^2 + 3x + 18}{(x-6)(x+6)} = \frac{(x+3)(x+6)}{(x-6)(x+6)}

Canceling Out Common Factors (continued)

We can see that $(x+6)$ is a common factor between the numerator and denominator, so we can cancel it out.

\frac{(x+3)(x+6)}{(x-6)(x+6)} = \frac{x+3}{x-6}

Conclusion

In this article, we simplified the expression: $\frac{x}{x-6} - \frac{3}{x+6}$ . We used various techniques, including finding a common denominator, combining fractions, and factoring, to simplify the expression. We also canceled out common factors between the numerator and denominator to obtain the final simplified expression: $\frac{x+3}{x-6}$ .

Final Answer

The final answer is $\boxed{\frac{x+3}{x-6}}$ .

Applications of Simplifying Algebraic Expressions

Simplifying algebraic expressions has numerous applications in various fields, including mathematics, physics, engineering, and computer science. Some of the applications of simplifying algebraic expressions include:

Solving equations and inequalities
Finding the maximum or minimum value of a function
Graphing functions
Solving systems of equations
Finding the area and perimeter of shapes
Finding the volume and surface area of solids

Tips and Tricks for Simplifying Algebraic Expressions

Here are some tips and tricks for simplifying algebraic expressions:

Use the distributive property to expand and combine like terms
Use the commutative and associative properties to rearrange terms
Use the identity property to simplify expressions
Use the inverse property to simplify expressions
Factor expressions to cancel out common factors
Use the quadratic formula to solve quadratic equations

Common Mistakes to Avoid When Simplifying Algebraic Expressions

Here are some common mistakes to avoid when simplifying algebraic expressions:

Not using the distributive property to expand and combine like terms
Not using the commutative and associative properties to rearrange terms
Not using the identity property to simplify expressions
Not using the inverse property to simplify expressions
Not factoring expressions to cancel out common factors
Not using the quadratic formula to solve quadratic equations

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we simplified the expression: $\frac{x}{x-6} - \frac{3}{x+6}$ . We used various techniques, including finding a common denominator, combining fractions, and factoring, to simplify the expression. We also canceled out common factors between the numerator and denominator to obtain the final simplified expression: $\frac{x+3}{x-6}$ .

Introduction

In our previous article, we simplified the expression: $\frac{x}{x-6} - \frac{3}{x+6}$ . We used various techniques, including finding a common denominator, combining fractions, and factoring, to simplify the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

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

A: The first step in simplifying an algebraic expression is to identify the type of expression and determine the best approach to simplify it. This may involve finding a common denominator, combining fractions, or factoring.

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

A: To find a common denominator for two fractions, you need to multiply the denominators together. For example, if you have two fractions with denominators of $x-6$ and $x+6$ , the common denominator would be $(x-6)(x+6)$ .

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

A: Combining fractions involves adding or subtracting the numerators of two or more fractions, while adding fractions involves adding the numerators of two or more fractions with the same denominator.

Q: How do I factor an algebraic expression?

A: To factor an algebraic expression, you need to identify the greatest common factor (GCF) of the terms and divide each term by the GCF. For example, if you have the expression $x^2 + 3x + 18$ , you can factor it as $(x+3)(x+6)$ .

Q: What is the purpose of canceling out common factors in an algebraic expression?

A: Canceling out common factors in an algebraic expression helps to simplify the expression and make it easier to work with. It also helps to eliminate any unnecessary terms and make the expression more manageable.

Q: How do I know when to use the distributive property to expand and combine like terms?

A: You should use the distributive property to expand and combine like terms when you have an expression with multiple terms and you need to simplify it. This involves multiplying each term by the other terms and combining like terms.

Q: What is the difference between the commutative and associative properties of addition and multiplication?

A: The commutative property of addition and multiplication states that the order of the terms does not change the result, while the associative property states that the order in which you perform the operations does not change the result.

Q: How do I use the identity property to simplify an algebraic expression?

A: You can use the identity property to simplify an algebraic expression by adding or subtracting zero to the expression. This helps to eliminate any unnecessary terms and make the expression more manageable.

Q: What is the purpose of using the inverse property to simplify an algebraic expression?

A: The inverse property is used to simplify an algebraic expression by adding or subtracting the additive inverse of a term. This helps to eliminate any unnecessary terms and make the expression more manageable.

Q: How do I use the quadratic formula to solve quadratic equations?

A: The quadratic formula is used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . It involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the equation.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has helped you to develop your problem-solving skills.

Final Tips and Tricks

Here are some final tips and tricks for simplifying algebraic expressions:

Practice, practice, practice: The more you practice simplifying algebraic expressions, the more comfortable you will become with the techniques and the more confident you will be in your ability to solve problems.
Use the distributive property to expand and combine like terms: This is a powerful technique for simplifying algebraic expressions and can help you to eliminate any unnecessary terms.
Use the commutative and associative properties to rearrange terms: This can help you to simplify the expression and make it easier to work with.
Use the identity property to simplify the expression: This can help you to eliminate any unnecessary terms and make the expression more manageable.
Use the inverse property to simplify the expression: This can help you to eliminate any unnecessary terms and make the expression more manageable.
Use the quadratic formula to solve quadratic equations: This is a powerful technique for solving quadratic equations and can help you to find the solutions to the equation.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying algebraic expressions:

Not using the distributive property to expand and combine like terms
Not using the commutative and associative properties to rearrange terms
Not using the identity property to simplify the expression
Not using the inverse property to simplify the expression
Not factoring expressions to cancel out common factors
Not using the quadratic formula to solve quadratic equations

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we answered some frequently asked questions (FAQs) related to simplifying algebraic expressions and provided some final tips and tricks for simplifying algebraic expressions. We hope that this article has provided you with a better understanding of how to simplify algebraic expressions and has helped you to develop your problem-solving skills.