Simplify The Expression -2(p+4)^2-3+5p A Step-by-Step Guide

In this article, we will delve into the process of simplifying the algebraic expression $-2(p+4)^2-3+5p$. This involves expanding, combining like terms, and arranging the expression in standard form. Understanding how to simplify algebraic expressions is a fundamental skill in mathematics, and this guide aims to provide a clear and comprehensive explanation of the steps involved. We will break down the problem into manageable parts, ensuring that you grasp each concept thoroughly. By the end of this article, you will not only be able to simplify this particular expression but also have a solid foundation for tackling similar problems.

Understanding the Basics of Algebraic Simplification

Before we dive into the specific problem, let's review some essential concepts related to algebraic simplification. Algebraic expressions are combinations of variables, constants, and mathematical operations. The goal of simplification is to rewrite an expression in a more concise and manageable form without changing its value. This often involves expanding brackets, combining like terms, and rearranging terms in a specific order. The standard form of a quadratic expression, which we will encounter in this problem, is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Understanding these basics is crucial for successfully simplifying the given expression.

Expanding the Squared Term

The first step in simplifying $-2(p+4)^2-3+5p$ is to expand the squared term $(p+4)^2$. To do this, we use the formula $(a+b)^2 = a^2 + 2ab + b^2$. Applying this formula to $(p+4)^2$, we get:

$(p+4)^2 = p^2 + 2(p)(4) + 4^2 = p^2 + 8p + 16$

This expansion is a critical step, as it transforms the squared term into a more manageable form that we can then multiply by the constant outside the parentheses. Mastering the expansion of squared terms is essential for simplifying many algebraic expressions. It allows us to remove the parentheses and combine the resulting terms with other terms in the expression. The correct expansion of $(p+4)^2$ lays the groundwork for the subsequent steps in the simplification process. This initial step is crucial for accurately simplifying the entire expression.

Distributing the Constant

Now that we have expanded $(p+4)^2$ to $p^2 + 8p + 16$, we need to multiply this expression by $-2$. This is done by distributing the $-2$ across each term inside the parentheses:

$-2(p^2 + 8p + 16) = -2p^2 - 16p - 32$

This step is crucial because it removes the parentheses and allows us to combine like terms later. Distribution is a fundamental operation in algebra, and it's important to perform it accurately. Each term inside the parentheses must be multiplied by the constant outside. This ensures that the value of the expression remains unchanged while we are simplifying it. Paying close attention to the signs (positive and negative) during distribution is particularly important to avoid errors. Correct distribution sets the stage for the next steps in simplifying the expression.

Combining Like Terms

After distributing the $-2$, our expression looks like this: $-2p^2 - 16p - 32 - 3 + 5p$. Now, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms with $p$ ($-16p$ and $5p$) and two constant terms ($-32$ and $-3$). Combining these, we get:

$-16p + 5p = -11p$

$-32 - 3 = -35$

So, our expression now becomes $-2p^2 - 11p - 35$. Combining like terms is a fundamental step in simplifying algebraic expressions. It reduces the number of terms and makes the expression more concise. Identifying and combining like terms correctly is essential for arriving at the final simplified form. This process involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged. The ability to combine like terms accurately is a cornerstone of algebraic manipulation.

Arranging in Standard Form

The final step is to arrange the expression in standard form. The standard form for a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, the expression is already in standard form: $-2p^2 - 11p - 35$.

Here, $a = -2$, $b = -11$, and $c = -35$. Arranging an expression in standard form is important because it makes it easier to identify the coefficients and constants, which is often necessary for further mathematical operations, such as factoring or solving equations. The standard form provides a consistent way of representing expressions, which facilitates communication and understanding in mathematics. Ensuring that the expression is in standard form is the final touch in the simplification process.

Step-by-Step Solution

Let's recap the steps we took to simplify the expression $-2(p+4)^2-3+5p$:

1. Expand the squared term: $(p+4)^2 = p^2 + 8p + 16$
2. Distribute the constant: $-2(p^2 + 8p + 16) = -2p^2 - 16p - 32$
3. Combine like terms: $-2p^2 - 16p - 32 - 3 + 5p = -2p^2 - 11p - 35$
4. Arrange in standard form: The expression is already in standard form: $-2p^2 - 11p - 35$

Therefore, the simplified expression in standard form is $-2p^2 - 11p - 35$. Following a step-by-step approach is crucial for simplifying algebraic expressions accurately. Each step builds upon the previous one, and careful execution of each step ensures that the final result is correct. This methodical approach not only helps in simplifying the expression but also enhances understanding of the underlying algebraic principles.

Common Mistakes to Avoid

When simplifying algebraic expressions, several common mistakes can occur. Here are a few to watch out for:

• Incorrectly expanding squared terms: For example, students might incorrectly state that $(p+4)^2 = p^2 + 16$ without including the middle term $2ab$. Accurate expansion of squared terms is essential. Remember the formula $(a+b)^2 = a^2 + 2ab + b^2$. Failing to include the $2ab$ term is a frequent error that can lead to an incorrect simplification.
• Errors in distribution: Forgetting to distribute the constant to all terms inside the parentheses is another common mistake. Ensure that every term inside the parentheses is multiplied by the constant outside. This can be avoided by systematically multiplying the constant by each term inside the parentheses, paying close attention to the signs.
• Incorrectly combining like terms: Mixing up terms with different powers of the variable or making arithmetic errors when adding or subtracting coefficients can lead to mistakes. Carefully identify and combine like terms, paying attention to both the variable part and the coefficients. This requires a methodical approach and a good understanding of what constitutes a like term.
• Forgetting to arrange in standard form: While not strictly a mistake in simplification, failing to arrange the expression in standard form can lead to confusion in subsequent steps. Always arrange the final expression in standard form $ax^2 + bx + c$. This ensures consistency and facilitates further mathematical operations.

By being aware of these common pitfalls, you can increase your accuracy and confidence in simplifying algebraic expressions.

Practice Problems

To solidify your understanding, let's look at a few practice problems:

1. Simplify: $3(x-2)^2 + 4x - 5$
2. Simplify: $-4(y+1)^2 - 2y + 7$
3. Simplify: $2(z-3)^2 + 6z - 1$

Working through these problems will help you apply the steps we've discussed and identify any areas where you might need further practice. Practice is key to mastering algebraic simplification. The more you practice, the more comfortable and confident you will become in applying the rules and procedures. These practice problems provide an opportunity to reinforce your understanding and develop your problem-solving skills.

Conclusion

Simplifying the expression $-2(p+4)^2-3+5p$ involves expanding, distributing, combining like terms, and arranging the result in standard form. The simplified expression is $-2p^2 - 11p - 35$. Mastering the simplification of algebraic expressions is a critical skill in algebra. It involves a series of steps that must be performed accurately to arrive at the correct result. By understanding the underlying principles and practicing regularly, you can develop proficiency in simplifying various types of expressions. This skill is not only essential for success in algebra but also forms the foundation for more advanced mathematical concepts.

By following the steps outlined in this guide and avoiding common mistakes, you can confidently simplify algebraic expressions. Remember, practice makes perfect, so keep working at it! Continuous practice and a solid understanding of the fundamentals are the keys to success in algebraic simplification. With consistent effort, you can develop the skills and confidence needed to tackle complex problems and excel in mathematics.