Simplifying The Expression -3(x+3)^2-3+3x A Step-by-Step Guide

Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to represent complex equations in a more manageable and understandable form. In this comprehensive guide, we will walk through the process of simplifying the expression $-3(x+3)^2-3+3x$, step by step. We will break down each part of the expression, explain the mathematical principles involved, and provide clear instructions on how to arrive at the simplified standard form. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with the knowledge and confidence to tackle similar problems.

Understanding the Importance of Simplifying Expressions

Before we dive into the specifics of simplifying the given expression, let's understand why simplification is so crucial in mathematics. Simplifying expressions makes them easier to work with, whether you're solving equations, graphing functions, or performing other mathematical operations. A simplified expression is more concise, which reduces the chances of making errors in subsequent calculations. It also helps in identifying the key characteristics of the expression, such as its degree, leading coefficient, and constant term. For instance, a quadratic expression in standard form ($ax^2 + bx + c$) immediately reveals the coefficients that determine the shape and position of the parabola when graphed.

Simplification is not just a mathematical exercise; it is a practical skill used in various fields, including physics, engineering, computer science, and economics. In physics, simplified expressions can help describe the motion of objects or the behavior of electrical circuits. In engineering, they are used to design structures and systems. In computer science, simplified algorithms can lead to more efficient code. In economics, they help in modeling market behavior and making predictions. Therefore, mastering the art of simplifying expressions is a valuable asset in both academic and professional settings. The ability to manipulate and simplify expressions efficiently is a cornerstone of problem-solving in these diverse fields.

Breaking Down the Expression: $-3(x+3)^2-3+3x$

To simplify the expression $-3(x+3)^2-3+3x$, we need to follow the order of operations (PEMDAS/BODMAS), which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order ensures that we perform operations in the correct sequence, leading to the correct simplified form. The given expression involves several operations, including exponentiation, multiplication, and addition/subtraction. We will tackle each operation step by step, ensuring we adhere to the order of operations to avoid errors.

Step 1: Expanding the Squared Term $(x+3)^2$

The first step in simplifying the expression is to expand the squared term $(x+3)^2$. This means multiplying $(x+3)$ by itself. We can use the distributive property (also known as the FOIL method) to do this:

$(x+3)^2 = (x+3)(x+3) = x(x) + x(3) + 3(x) + 3(3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9$

Expanding the squared term is a critical step because it converts the expression inside the parentheses into a polynomial form that can be further simplified. The distributive property is a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. By correctly applying the distributive property, we ensure that each term in the first set of parentheses is multiplied by each term in the second set, leading to the correct expanded form of the expression. This step lays the groundwork for the subsequent operations, as it removes the exponent and allows us to combine like terms in the following steps.

Step 2: Multiplying by $-3$

Now that we have expanded $(x+3)^2$ to $x^2 + 6x + 9$, we need to multiply this expression by $-3$:

$-3(x^2 + 6x + 9) = -3(x^2) + (-3)(6x) + (-3)(9) = -3x^2 - 18x - 27$

This step involves the distributive property again, where we multiply each term inside the parentheses by $-3$. It is crucial to pay attention to the signs while performing this multiplication. Multiplying by a negative number changes the sign of each term inside the parentheses. This step is important because it eliminates the parentheses and prepares the expression for the final simplification steps. Accurate multiplication is essential to avoid errors that could propagate through the rest of the simplification process. The result of this step is a quadratic expression that is now ready to be combined with the remaining terms in the original expression.

Step 3: Incorporating the Remaining Terms

Next, we bring in the remaining terms from the original expression: $-3$ and $+3x$. We now have:

$-3x^2 - 18x - 27 - 3 + 3x$

This step involves simply adding the remaining terms to the expression we obtained in the previous step. It is a straightforward process but is crucial for maintaining the integrity of the expression. Ensuring that all terms are correctly included is vital for arriving at the correct final answer. This step sets the stage for combining like terms, which is the final step in simplifying the expression.

Step 4: Combining Like Terms

The final step in simplifying the expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are the $x$ terms $(-18x$ and $+3x)$ and the constant terms $(-27$ and $-3)$. Combining these terms, we get:

$-3x^2 + (-18x + 3x) + (-27 - 3) = -3x^2 - 15x - 30$

Combining like terms is a fundamental algebraic operation that simplifies the expression by reducing the number of terms. This step involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged. Paying close attention to the signs of the coefficients is crucial for accurate simplification. By combining like terms, we arrive at the simplest form of the expression, which is easier to understand and work with in subsequent mathematical operations. This step completes the simplification process and provides us with the expression in its standard form.

The Simplified Expression in Standard Form

After following all the steps, we have simplified the expression $-3(x+3)^2-3+3x$ to:

$-3x^2 - 15x - 30$

This is the simplified expression in standard form, which means it is written with the terms in descending order of their exponents. The standard form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, $a = -3$, $b = -15$, and $c = -30$. The standard form allows us to easily identify the coefficients and the constant term, which are important for further analysis and operations, such as solving quadratic equations or graphing quadratic functions. The simplified form is not only more concise but also provides a clear representation of the expression's characteristics, making it easier to work with in various mathematical contexts.

Conclusion: Mastering Algebraic Simplification

In this guide, we have demonstrated the step-by-step process of simplifying the expression $-3(x+3)^2-3+3x$. We started by understanding the importance of simplification and then broke down the expression into manageable parts. We expanded the squared term, multiplied by the coefficient, incorporated the remaining terms, and finally combined like terms to arrive at the simplified expression in standard form: $-3x^2 - 15x - 30$.

Simplifying algebraic expressions is a fundamental skill in mathematics that requires a solid understanding of the order of operations and algebraic properties. By following a systematic approach and paying attention to detail, you can confidently simplify complex expressions and solve a wide range of mathematical problems. Practice is key to mastering this skill, so we encourage you to tackle more simplification problems to reinforce your understanding. With consistent effort and a clear understanding of the principles involved, you can become proficient in algebraic simplification and enhance your mathematical abilities.

Key Takeaways:

• Simplifying expressions makes them easier to work with and reduces the chances of errors.
• The order of operations (PEMDAS/BODMAS) is crucial for correct simplification.
• The distributive property is used to expand expressions and multiply terms.
• Combining like terms is the final step in simplification.
• The standard form of a quadratic expression is $ax^2 + bx + c$.

By mastering the techniques and principles outlined in this guide, you will be well-equipped to tackle more complex algebraic problems and excel in your mathematical studies. Keep practicing, and you'll find that simplifying expressions becomes second nature.