Simplify $3x(x - 12x) + 3x^2 - 2(x - 2)^2$: Steps & Truth

Hey math enthusiasts! Let's break down how to simplify the algebraic expression $3x(x - 12x) + 3x^2 - 2(x - 2)^2$. We'll walk through the process step by step and then identify some true statements about the simplification. Ready to dive in?

Understanding the Initial Expression

Before we start simplifying, it's crucial to understand what we're working with. The expression is $3x(x - 12x) + 3x^2 - 2(x - 2)^2$. It looks a bit intimidating at first, but we can tackle it by breaking it down into smaller, more manageable parts. Our main goal is to combine like terms and reduce the expression to its simplest form. This involves several algebraic techniques, including distribution, the order of operations (PEMDAS/BODMAS), and careful handling of signs. So, buckle up, because we're about to embark on a journey of algebraic simplification!

Step-by-Step Simplification Process

Let's meticulously walk through the simplification process. Each step will build upon the previous one, ensuring we arrive at the correct final answer. This detailed breakdown will not only help you understand the solution but also equip you with the skills to tackle similar problems in the future.

1. Simplify inside the parentheses:

   • We begin by simplifying the term $(x - 12x)$ inside the first set of parentheses.
   • Combining these like terms gives us $-11x$. So, our expression now becomes $3x(-11x) + 3x^2 - 2(x - 2)^2$.

2. Distribute the $3x$:

   • Next, we distribute the $3x$ across the $-11x$ inside the first term.
   • Multiplying $3x$ by $-11x$ results in $-33x^2$. Our expression is now $-33x^2 + 3x^2 - 2(x - 2)^2$.

3. Expand the squared binomial:

   • Here's where things get a little more interesting. We need to expand the term $(x - 2)^2$. Remember that squaring a binomial means multiplying it by itself: $(x - 2)(x - 2)$.
   • Using the FOIL (First, Outer, Inner, Last) method or the distributive property, we get:
     • $(x - 2)(x - 2) = x^2 - 2x - 2x + 4 = x^2 - 4x + 4$.
   • So, our expression becomes $-33x^2 + 3x^2 - 2(x^2 - 4x + 4)$.

4. Distribute the $-2$:

   • Now, we distribute the $-2$ across the expanded binomial $(x^2 - 4x + 4)$.
   • This gives us $-2x^2 + 8x - 8$.
   • Our expression is now $-33x^2 + 3x^2 - 2x^2 + 8x - 8$.

5. Combine like terms:

   • Finally, we combine all the like terms. We have $x^2$ terms and a single $x$ term, along with a constant term.
   • Combining the $x^2$ terms: $-33x^2 + 3x^2 - 2x^2 = -32x^2$.
   • Our simplified expression is $-32x^2 + 8x - 8$.

Key Takeaways from the Simplification Process

This step-by-step process highlights several important algebraic techniques. First, we tackled expressions within parentheses, simplifying them before moving on. Then, we used the distributive property to multiply terms across parentheses. A critical step was expanding the squared binomial, which required careful application of the FOIL method or the distributive property. Finally, we combined like terms to arrive at the most simplified form of the expression. This systematic approach is key to successfully simplifying complex algebraic expressions.

Analyzing the Statements About the Process and Simplified Product

Now that we've simplified the expression, let's evaluate the statements about the process and the simplified product. This is where we put our understanding of the simplification steps and the final expression to the test. We need to carefully examine each statement and determine whether it accurately reflects what we did and what we obtained.

Evaluating Statement A

Statement A: The term $-2(x - 2)^2$ is simplified by first squaring the expression $x - 2$.

This statement is true. As we saw in step 3 of our simplification process, the first thing we did with the term $-2(x - 2)^2$ was to expand the squared binomial $(x - 2)^2$. This involved multiplying $(x - 2)$ by itself, which is the definition of squaring an expression. So, statement A accurately describes our initial step in simplifying this part of the expression.

Considering Statement B (Hypothetical)

Let's imagine a statement B, for example: "The simplified product is a cubic expression." To evaluate such a statement, we would look at our final simplified expression, $-32x^2 + 8x - 8$. A cubic expression has a term with $x^3$, but our expression only has terms with $x^2$, $x$, and a constant. Therefore, this hypothetical statement B would be false.

Understanding the Simplified Product

Our simplified product, $-32x^2 + 8x - 8$, is a quadratic expression. This is because the highest power of $x$ in the expression is 2. The coefficients are -32, 8, and -8, representing the values multiplying the $x^2$, $x$, and constant terms, respectively. Understanding the nature of the simplified product helps us classify it and make further observations or manipulations if needed.

Selecting the True Options

Based on our step-by-step simplification and analysis, we can now select the true options. Remember, we're looking for three statements that accurately describe the process and the final simplified product.

Choosing the Correct Statements

After our thorough analysis, we can confidently select the true statements. The key is to match the statements with the specific steps we took and the characteristics of our final expression. Let's make sure our choices are well-supported by our work.

1. Statement A: As we've already established, this statement is true because we did indeed start by squaring the expression $(x - 2)$.
2. We need to identify two more correct statements based on the context provided (which was hypothetical for Statement B). These statements should reflect other aspects of the simplification process or the nature of the final simplified expression.

Guys, remember that simplifying algebraic expressions is like solving a puzzle. Each step needs to be logical and build upon the previous one. By breaking down the problem into smaller parts and carefully applying the rules of algebra, you can conquer even the most intimidating expressions! Keep practicing, and you'll become a simplification pro in no time!