Simplify The Algebraic Expression: $-2(x+3)(x-3)-(3x-2)^2$

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Understanding and Simplifying Algebraic Expressions

In the realm of mathematics, particularly in algebra, simplifying expressions is a fundamental skill. This process involves manipulating algebraic expressions to reduce them to their simplest form, making them easier to understand and work with. Simplifying expressions often involves combining like terms, applying the distributive property, and using algebraic identities. In this article, we will delve into the process of simplifying a complex algebraic expression, breaking down each step to ensure clarity and understanding.

The expression we aim to simplify is: $-2(x+3)(x-3)-(3x-2)^2$. This expression involves multiple operations, including multiplication of binomials, squaring a binomial, and distribution of constants. To effectively simplify this, we will employ the order of operations (PEMDAS/BODMAS) and utilize algebraic identities such as the difference of squares and the square of a binomial.

Step 1: Expanding the Difference of Squares

The first part of the expression, $-2(x+3)(x-3)$, involves the product of two binomials in the form of $(a+b)(a-b)$, which is a classic example of the difference of squares. The difference of squares identity states that $(a+b)(a-b) = a^2 - b^2$. Applying this identity to our expression, we can simplify $(x+3)(x-3)$ as follows:

$(x+3)(x-3) = x^2 - 3^2 = x^2 - 9$

Now, we substitute this back into the original expression:

$-2(x+3)(x-3) = -2(x^2 - 9)$

Next, we distribute the -2 across the parentheses:

$-2(x^2 - 9) = -2x^2 + 18$

This completes the simplification of the first part of the expression. The key takeaway here is the application of the difference of squares identity, which significantly simplifies the multiplication of the two binomials. Understanding and recognizing such identities is crucial for efficient algebraic manipulation. Furthermore, the proper application of the distributive property ensures that each term inside the parentheses is correctly multiplied by the constant outside.

Step 2: Expanding the Square of a Binomial

The second part of the expression, $-(3x-2)^2$, involves squaring a binomial. The square of a binomial identity states that $(a-b)^2 = a^2 - 2ab + b^2$. Applying this identity to $(3x-2)^2$, we get:

$(3x-2)^2 = (3x)^2 - 2(3x)(2) + (2)^2$

Now, we perform the individual operations:

$(3x)^2 = 9x^2$

$2(3x)(2) = 12x$

$(2)^2 = 4$

So, $(3x-2)^2 = 9x^2 - 12x + 4$

However, we must not forget the negative sign in front of the term in the original expression. Therefore, we need to distribute the negative sign across the entire expanded binomial:

$-(3x-2)^2 = -(9x^2 - 12x + 4) = -9x^2 + 12x - 4$

This step highlights the importance of careful application of the binomial square identity and the subsequent distribution of the negative sign. Errors in this step can lead to an incorrect final simplified expression. Paying close attention to signs and ensuring each term is correctly accounted for is vital.

Step 3: Combining Like Terms

Now that we have simplified both parts of the expression, we can combine them:

$-2(x+3)(x-3) - (3x-2)^2 = (-2x^2 + 18) + (-9x^2 + 12x - 4)$

To combine like terms, we group terms with the same variable and exponent together:

$(-2x^2 - 9x^2) + (12x) + (18 - 4)$

Now, we perform the addition and subtraction:

$-11x^2 + 12x + 14$

Thus, the simplified expression is $-11x^2 + 12x + 14$. This final step brings together the results of the previous steps, combining like terms to arrive at the most simplified form of the expression. The ability to accurately combine like terms is a fundamental skill in algebra and is crucial for simplifying more complex expressions.

Detailed Breakdown of the Simplification Process

To further illustrate the process, let's break down each step with detailed explanations and justifications. This will help in understanding the underlying principles and techniques used in simplifying algebraic expressions.

Initial Expression

The initial expression we are working with is:

$-2(x+3)(x-3)-(3x-2)^2$

This expression involves several algebraic operations and requires a systematic approach to simplify it effectively.

Expanding $-2(x+3)(x-3)$

As discussed earlier, this part of the expression can be simplified using the difference of squares identity. Let's go through the steps again:

1. Recognize the Difference of Squares: The term $(x+3)(x-3)$ is in the form of $(a+b)(a-b)$, where $a = x$ and $b = 3$.
2. Apply the Identity: Using the identity $(a+b)(a-b) = a^2 - b^2$, we get $(x+3)(x-3) = x^2 - 3^2 = x^2 - 9$.
3. Distribute -2: Now, we multiply the result by -2: $-2(x^2 - 9) = -2x^2 + 18$.

This simplification is crucial because it reduces the complexity of the expression and makes it easier to combine with other terms later on.

Expanding $-(3x-2)^2$

This part involves squaring a binomial and then applying a negative sign. Let's break it down:

1. Recognize the Square of a Binomial: The term $(3x-2)^2$ is in the form of $(a-b)^2$, where $a = 3x$ and $b = 2$.
2. Apply the Identity: Using the identity $(a-b)^2 = a^2 - 2ab + b^2$, we get $(3x-2)^2 = (3x)^2 - 2(3x)(2) + (2)^2$.
3. Simplify Each Term:
   • $(3x)^2 = 9x^2$
   • $2(3x)(2) = 12x$
   • $(2)^2 = 4$
4. Combine the Terms: So, $(3x-2)^2 = 9x^2 - 12x + 4$.
5. Distribute the Negative Sign: Now, we apply the negative sign: $-(9x^2 - 12x + 4) = -9x^2 + 12x - 4$.

This step requires careful attention to detail, particularly when dealing with the negative sign. Ensuring that the negative sign is correctly distributed across all terms is essential for accuracy.

Combining Like Terms

Now that we have simplified both parts of the expression, we can combine like terms:

1. Write Down the Simplified Parts:
   • $-2(x+3)(x-3) = -2x^2 + 18$
   • $-(3x-2)^2 = -9x^2 + 12x - 4$
2. Combine the Expressions: $-2x^2 + 18 - 9x^2 + 12x - 4$
3. Group Like Terms: $(-2x^2 - 9x^2) + (12x) + (18 - 4)$
4. Perform the Operations:
   • $-2x^2 - 9x^2 = -11x^2$
   • $18 - 4 = 14$
5. Write the Final Simplified Expression: $-11x^2 + 12x + 14$

Common Mistakes to Avoid

When simplifying algebraic expressions, several common mistakes can occur. Being aware of these pitfalls can help in avoiding them and ensuring accurate simplification.

1. Incorrectly Applying the Distributive Property: A common mistake is failing to distribute a constant or a negative sign across all terms inside parentheses. For example, in the expression $-(3x-2)^2$, it's crucial to distribute the negative sign after expanding the binomial.
2. Misusing Algebraic Identities: Algebraic identities like the difference of squares and the square of a binomial are powerful tools, but they must be applied correctly. Confusing the identities or misapplying them can lead to incorrect simplifications.
3. Combining Unlike Terms: Only like terms (terms with the same variable and exponent) can be combined. A common mistake is adding or subtracting terms that are not like terms.
4. Sign Errors: Sign errors are a frequent cause of mistakes in algebraic simplification. Pay close attention to the signs of each term, especially when distributing negative signs or combining terms.
5. Order of Operations: Failing to follow the correct order of operations (PEMDAS/BODMAS) can lead to errors. Ensure that exponents are handled before multiplication and division, and addition and subtraction are performed last.

Real-World Applications of Simplifying Expressions

Simplifying algebraic expressions is not just an academic exercise; it has numerous real-world applications. These applications span various fields, including engineering, physics, economics, and computer science.

1. Engineering: In engineering, simplified expressions are used to model physical systems, design structures, and analyze circuits. Simplifying equations allows engineers to make accurate predictions and optimize designs.
2. Physics: Physics relies heavily on mathematical models to describe natural phenomena. Simplifying expressions is crucial for solving physics problems, such as calculating motion, forces, and energy.
3. Economics: Economic models often involve complex equations. Simplifying these equations helps economists analyze market trends, make predictions, and develop policies.
4. Computer Science: In computer science, simplified expressions are used in algorithm design, data analysis, and optimization problems. Simplifying code and mathematical expressions can improve efficiency and performance.
5. Financial Analysis: Financial analysts use algebraic expressions to model investments, calculate returns, and assess risk. Simplifying these expressions helps in making informed financial decisions.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics with wide-ranging applications. In this article, we have demonstrated the process of simplifying the expression $-2(x+3)(x-3)-(3x-2)^2$ step by step. We began by expanding the difference of squares, then expanded the square of a binomial, and finally combined like terms to arrive at the simplified expression $-11x^2 + 12x + 14$. Along the way, we highlighted the importance of understanding and applying algebraic identities, avoiding common mistakes, and recognizing the real-world applications of simplifying expressions. Mastering these techniques will enhance your algebraic skills and enable you to tackle more complex mathematical problems with confidence. Remember, practice is key to proficiency in mathematics, so continue to work through various examples to solidify your understanding. The ability to simplify expressions efficiently is a valuable asset in both academic and professional settings, empowering you to solve problems and make informed decisions across a variety of disciplines.