Simplifying Expressions A Step-by-Step Guide To -3(y-5)^2-9+7y

This article provides a comprehensive walkthrough of simplifying the algebraic expression $-3(y-5)^2-9+7y$. We will delve into the order of operations, common mistakes to avoid, and the correct steps to arrive at the simplified form. Understanding how to simplify such expressions is crucial for success in algebra and beyond. Let's embark on this journey to unravel the complexities of this expression and master the art of algebraic simplification.

Understanding the Order of Operations

When simplifying algebraic expressions, it's crucial to adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations must be performed to ensure accurate results. In the context of our expression, $-3(y-5)^2-9+7y$, following PEMDAS is paramount.

The first step involves addressing the parentheses. Within the parentheses, we have $(y-5)$, which cannot be simplified further at this stage as it involves a variable and a constant. The next operation according to PEMDAS is dealing with the exponent. We have $(y-5)^2$, which means we need to square the binomial $(y-5)$. This involves multiplying the binomial by itself: $(y-5)(y-5)$. Using the FOIL method (First, Outer, Inner, Last) or the distributive property, we expand this to $y^2 - 5y - 5y + 25$, which simplifies to $y^2 - 10y + 25$. This step is critical as it sets the stage for further simplification by removing the exponent and expressing the term as a quadratic expression. Failing to correctly expand the squared binomial is a common error that can lead to an incorrect final answer. Therefore, meticulous attention to detail is necessary when handling exponents in algebraic expressions.

Step-by-Step Simplification of $-3(y-5)^2-9+7y$

To effectively simplify the expression $-3(y-5)^2-9+7y$, we must meticulously follow the order of operations (PEMDAS). This ensures we arrive at the correct simplified form. Let's break down the process step by step.

Step 1: Expanding the Squared Binomial

The initial focus is on the term $(y-5)^2$. This represents the square of a binomial, and to expand it correctly, we multiply the binomial by itself: $(y-5)(y-5)$. Applying the FOIL method (First, Outer, Inner, Last) or the distributive property, we get:

• First: $y * y = y^2$
• Outer: $y * -5 = -5y$
• Inner: $-5 * y = -5y$
• Last: $-5 * -5 = 25$

Combining these terms, we have $y^2 - 5y - 5y + 25$. Simplifying further by combining like terms (-5y and -5y), we arrive at $y^2 - 10y + 25$. This quadratic expression is the expanded form of the squared binomial and is a crucial intermediate result.

Step 2: Distributing the -3

Now that we've expanded $(y-5)^2$ to $y^2 - 10y + 25$, the next step is to distribute the -3 across the terms within the parentheses. This means multiplying each term inside the parentheses by -3:

• $-3 * y^2 = -3y^2$
• $-3 * -10y = 30y$
• $-3 * 25 = -75$

This gives us the expression $-3y^2 + 30y - 75$. Distributing the negative sign correctly is critical, as it affects the sign of each term inside the parentheses. A common mistake is to forget to distribute the negative sign to all terms, which can lead to an incorrect simplification.

Step 3: Combining Like Terms

After distributing the -3, our expression looks like this: $-3y^2 + 30y - 75 - 9 + 7y$. The next step is to identify and combine like terms. Like terms are those that have the same variable raised to the same power. In this expression, we have two types of like terms: terms with the variable 'y' and constant terms.

• Terms with 'y': We have $30y$ and $7y$. Combining these gives us $30y + 7y = 37y$.
• Constant terms: We have $-75$ and $-9$. Combining these gives us $-75 - 9 = -84$.

Now, we substitute these combined terms back into the expression. The $-3y^2$ term remains unchanged as it has no like terms to combine with.

Step 4: Writing the Simplified Expression

After combining like terms, we can now write the simplified expression. We have the following terms:

• $-3y^2$ (quadratic term)
• $37y$ (linear term)
• $-84$ (constant term)

Combining these, the simplified expression is $-3y^2 + 37y - 84$. This is the final simplified form of the original expression, $-3(y-5)^2-9+7y$. By following the order of operations and meticulously combining like terms, we have successfully simplified the algebraic expression.

Common Mistakes to Avoid

Simplifying algebraic expressions can be challenging, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate simplification. Here are some key mistakes to watch out for when simplifying expressions like $-3(y-5)^2-9+7y$:

Incorrectly Expanding the Squared Binomial

One of the most frequent errors occurs when expanding the squared binomial, $(y-5)^2$. Students may mistakenly apply the square to each term inside the parentheses, writing $y^2 - 25$, which is incorrect. The correct method, as discussed earlier, involves multiplying the binomial by itself: $(y-5)(y-5)$. Using the FOIL method or distributive property yields $y^2 - 10y + 25$. Failing to recognize and correctly apply this expansion can lead to significant errors in the final simplified expression.

Distributing the Negative Sign Incorrectly

Another common mistake arises when distributing the -3 across the terms inside the parentheses after expanding the binomial. It's essential to multiply each term by -3, paying close attention to the signs. For instance, $-3 * -10y$ should result in $+30y$, not $-30y$. Similarly, $-3 * 25$ should be $-75$. Forgetting to distribute the negative sign to all terms or making sign errors during distribution can lead to an incorrect expression.

Combining Unlike Terms

A fundamental rule in algebra is that only like terms can be combined. Like terms have the same variable raised to the same power. For example, $30y$ and $7y$ are like terms and can be combined, but $30y$ and $-3y^2$ are not because they have different powers of 'y'. A common error is to mistakenly combine unlike terms, which results in an incorrect simplified expression. Always ensure that terms have the same variable and exponent before combining them.

Ignoring the Order of Operations (PEMDAS)

The order of operations, PEMDAS, is crucial for correct simplification. Neglecting this order can lead to significant errors. For example, in the expression $-3(y-5)^2-9+7y$, the exponent must be addressed before multiplication and addition. Failing to square the binomial $(y-5)$ before distributing the -3 will result in an incorrect simplification. Always adhere to PEMDAS to ensure the correct sequence of operations.

Arithmetic Errors

Simple arithmetic mistakes can also derail the simplification process. These can include errors in multiplication, addition, or subtraction. For instance, incorrectly multiplying -3 by 25 or making a mistake when combining -75 and -9 can lead to a wrong final answer. Double-checking each arithmetic operation can help minimize these errors.

Conclusion

In conclusion, simplifying the algebraic expression $-3(y-5)^2-9+7y$ requires a systematic approach, a thorough understanding of the order of operations, and careful attention to detail. By correctly expanding the squared binomial, distributing the -3, combining like terms, and avoiding common mistakes, we arrive at the simplified form: $-3y^2 + 37y - 84$. This process underscores the importance of mastering fundamental algebraic techniques for tackling more complex mathematical problems. Remember to always double-check your work and be mindful of potential errors to ensure accuracy in your simplifications.