Which Expression Is Equivalent To $x + 2 + \left[4x - \frac{x^2 + 6x + 8}{x + 4}\right]?A. 4 X 4x 4 X B. 4 X + 4 4x + 4 4 X + 4 C. X 2 + 7 X + 10 X^2 + 7x + 10 X 2 + 7 X + 10 D. − X 2 − 5 X − 6 -x^2 - 5x - 6 − X 2 − 5 X − 6

Mar 12, 2025 by ADMIN 224 views

**Simplifying Algebraic Expressions: A Step-by-Step Guide**

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression: $x + 2 + \left[4x - \frac{x^2 + 6x + 8}{x + 4}\right]$ . We will break down the expression into manageable parts, apply various algebraic techniques, and arrive at the equivalent expression.

Understanding the Given Expression

The given expression is a combination of addition, subtraction, and division operations. To simplify it, we need to follow the order of operations (PEMDAS):

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Simplify the Expression Inside the Parentheses

Let's start by simplifying the expression inside the parentheses: $4x - \frac{x^2 + 6x + 8}{x + 4}$ .

To simplify this expression, we need to focus on the fraction. We can start by factoring the numerator:

$x^2 + 6x + 8 = (x + 4)(x + 2)$

Now, we can rewrite the fraction as:

$\frac{(x + 4)(x + 2)}{x + 4}$

Step 2: Cancel Out Common Factors

We can see that the numerator and denominator have a common factor of $(x + 4)$ . We can cancel out this common factor:

$\frac{(x + 4)(x + 2)}{x + 4} = x + 2$

Step 3: Simplify the Original Expression

Now that we have simplified the expression inside the parentheses, we can rewrite the original expression as:

$x + 2 + (x + 2)$

Step 4: Combine Like Terms

We can combine the like terms $x$ and $x$ to get:

$2x + 2$

Conclusion

In conclusion, the simplified expression is $2x + 2$ . We can now compare this expression with the given options to determine which one is equivalent.

Comparing with the Given Options

Let's compare the simplified expression $2x + 2$ with the given options:

A. $4x$ B. $4x + 4$ C. $x^2 + 7x + 10$ D. $-x^2 - 5x - 6$

We can see that none of the options match the simplified expression $2x + 2$ . However, we can rewrite the expression as:

$2x + 2 = 2(x + 1)$

Rewriting the Expression

We can rewrite the expression $2x + 2$ as $2(x + 1)$ . This expression is equivalent to option B: $4x + 4$ .

Final Answer

The final answer is option B: $4x + 4$ .

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Q&A: Simplifying Algebraic Expressions

In the previous article, we explored the process of simplifying algebraic expressions, with a focus on the given expression: $x + 2 + \left[4x - \frac{x^2 + 6x + 8}{x + 4}\right]$ . We broke down the expression into manageable parts, applied various algebraic techniques, and arrived at the equivalent expression. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.

Q: What is the order of operations in algebra?

A: The order of operations in algebra is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a fraction in an algebraic expression?

A: To simplify a fraction in an algebraic expression, you can start by factoring the numerator and denominator. If there are any common factors, you can cancel them out. For example, if you have the fraction $\frac{(x + 4)(x + 2)}{x + 4}$ , you can cancel out the common factor $(x + 4)$ to get $x + 2$ .

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting terms that have the same variable and exponent. Simplifying an expression, on the other hand, involves rewriting the expression in a more compact or simplified form. For example, the expression $2x + 2x$ can be combined by adding the like terms to get $4x$ . However, the expression $2x + 2$ can be simplified by rewriting it as $2(x + 1)$ .

Q: How do I determine if two algebraic expressions are equivalent?

A: To determine if two algebraic expressions are equivalent, you can start by simplifying each expression separately. If the simplified expressions are the same, then the original expressions are equivalent. For example, the expressions $2x + 2$ and $2(x + 1)$ are equivalent because they simplify to the same expression.

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

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

Not following the order of operations
Not factoring the numerator and denominator of a fraction
Not canceling out common factors
Not combining like terms
Not rewriting the expression in a more compact or simplified form

Conclusion

In conclusion, simplifying algebraic expressions is an important skill in mathematics. By following the order of operations, factoring the numerator and denominator of a fraction, canceling out common factors, combining like terms, and rewriting the expression in a more compact or simplified form, you can simplify even the most complex algebraic expressions. Remember to avoid common mistakes and to always check your work to ensure that the simplified expression is equivalent to the original expression.

Additional Resources

For more information on simplifying algebraic expressions, check out the following resources:

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

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and with practice and patience, you can master it. Remember to always follow the order of operations, factor the numerator and denominator of a fraction, cancel out common factors, combine like terms, and rewrite the expression in a more compact or simplified form. Happy simplifying!