Simplify The Algebraic Expression: $2x(x-6)-7x^2-(13x-3)$

Hey guys! Today we're diving deep into the world of algebra to tackle a classic problem: simplifying a complex expression. If you've ever found yourself staring at a jumble of variables and numbers and wondering where to even begin, you're in the right place. We're going to break down the expression $2 x(x-6)-7 x^2-(13 x-3)$ step-by-step, making sure you understand each move. By the end of this, you'll be a pro at simplifying expressions and ready to conquer any algebraic challenge that comes your way. So, grab your notebooks, and let's get started on unraveling this mathematical puzzle!

Unpacking the Expression: The First Steps

Alright, let's start by looking at the expression we need to simplify: $2 x(x-6)-7 x^2-(13 x-3)$. Our main goal here is to combine all the like terms and get the expression into its most basic, or simplest form. Think of it like tidying up a messy room; we want to put all the similar items together. The first part we need to deal with is the $2x(x-6)$ term. This involves the distributive property, which means we multiply the $2x$ by each term inside the parentheses. So, $2x$ times $x$ gives us $2x^2$, and $2x$ times $-6$ gives us $-12x$. This transforms the first part of our expression into $2x^2 - 12x$. Now our expression looks like this: $2x^2 - 12x - 7x^2 - (13x - 3)$. We're already making progress, guys!

Tackling Parentheses and Negative Signs

Next up, we have to deal with the term $-(13x - 3)$. This negative sign in front of the parentheses is super important, and it's a common spot where mistakes happen. When you have a negative sign outside parentheses, it's like distributing a $-1$ to everything inside. So, the $-1$ times $13x$ becomes $-13x$, and the $-1$ times $-3$ becomes $+3$. This is crucial: a negative times a negative equals a positive! So, the $-(13x - 3)$ part simplifies to $-13x + 3$. Our expression is now getting a lot cleaner: $2x^2 - 12x - 7x^2 - 13x + 3$. We've successfully removed all the parentheses and are ready for the next big step: combining our like terms. Keep up the great work!

Combining Like Terms: The Final Push

Now that we've distributed and handled those pesky negative signs, we're left with $2x^2 - 12x - 7x^2 - 13x + 3$. The next logical step, and the one that will get us to the simplest form, is to combine our like terms. Like terms are terms that have the exact same variable raised to the exact same power. In our expression, the $x^2$ terms are $2x^2$ and $-7x^2$. Let's combine them: $2x^2 - 7x^2 = (2-7)x^2 = -5x^2$. See? We're just adding or subtracting the coefficients (the numbers in front of the variables). Now, let's look at the $x$ terms: $-12x$ and $-13x$. Combining these gives us $-12x - 13x = (-12-13)x = -25x$. Finally, we have our constant term, which is just the number without any variables: $+3$. So, putting it all together, our simplified expression is $-5x^2 - 25x + 3$. This is our final answer, the simplest form of the original expression. Pretty cool, right?

Verifying the Simplest Form and Understanding the Options

We've worked hard to simplify the expression $2 x(x-6)-7 x^2-(13 x-3)$ and arrived at $-5x^2 - 25x + 3$. Now, it's always a good idea to double-check our work, especially when dealing with multiple steps and signs. Let's quickly recap the process: first, we distributed the $2x$ to get $2x^2 - 12x$. Then, we distributed the negative sign to the second set of parentheses to get $-13x + 3$. Our expression became $2x^2 - 12x - 7x^2 - 13x + 3$. Finally, we combined the $x^2$ terms ($2x^2 - 7x^2 = -5x^2$), the $x$ terms ($-12x - 13x = -25x$), and the constant term ($+3$). This indeed gives us $-5x^2 - 25x + 3$. Now let's look at the multiple-choice options provided:

A. $5 x^2-25 x+3$ B. $-5 x^2-25 x+3$ C. $-5 x^2+x+3$ D. $-5 x^2-25 x-3$

Comparing our result, $-5x^2 - 25x + 3$, with the options, we can clearly see that Option B matches our answer perfectly. This confirms that our step-by-step simplification was accurate. It’s vital to be careful with signs and the order of operations when simplifying algebraic expressions to avoid common errors like mixing up positive and negative results or incorrectly combining terms. Keep practicing, and you'll become a simplification ninja in no time!

Key Concepts in Simplification

To truly master simplifying algebraic expressions like the one we just tackled, it's essential to have a solid grip on a few core mathematical concepts. These aren't just random rules; they are the building blocks that allow us to manipulate and understand equations. The first, and perhaps most fundamental, is the distributive property. We used this when we multiplied $2x$ by $(x-6)$, effectively distributing the $2x$ to both the $x$ and the $-6$. Remember, $a(b+c) = ab + ac$. This property is your best friend when dealing with parentheses. It allows you to expand expressions and remove those brackets, which is often the first step toward combining terms.

The Power of Combining Like Terms

Following the distributive property, the next crucial skill is combining like terms. This is where we group similar algebraic terms together to simplify the expression. Like terms are those that have the identical variable(s) raised to the identical power(s). For instance, in $3x^2 + 5x - 2x^2 + 7$, the terms $3x^2$ and $-2x^2$ are like terms because they both contain $x^2$. Similarly, $5x$ is a term, and $7$ is a constant term. When we combine them, we add or subtract their coefficients. So, $3x^2 - 2x^2$ becomes $1x^2$ or just $x^2$, and $5x$ stays as it is, and $7$ is our constant. The expression then simplifies to $x^2 + 5x + 7$. This process significantly reduces the complexity of an expression, making it easier to analyze and work with. Understanding how to identify and combine these terms is critical for solving equations and performing more advanced algebraic operations.

Order of Operations and Sign Rules

Beyond distribution and combining terms, it's also vital to pay close attention to the order of operations (often remembered by PEMDAS/BODMAS) and sign rules. The order of operations dictates the sequence in which you perform calculations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our example, we dealt with parentheses first by distributing. The sign rules, especially when dealing with negative signs, are just as important. Remember that multiplying or dividing two negatives results in a positive (e.g., $-(-3) = 3$, $-3 imes -4 = 12$), while multiplying or dividing a positive and a negative results in a negative (e.g., $-(+3) = -3$, $-3 imes 4 = -12$). These fundamental rules are the backbone of accurate algebraic manipulation. Mastering them ensures that you can confidently navigate through complex expressions and arrive at the correct, simplest form every single time. Keep these concepts in your toolkit, and you'll be well on your way to algebraic success!