Simplify $-3(y+2)^2-5+6y$ In Standard Form

Hey guys! Today, we're diving into a math problem where we need to simplify a quadratic expression. It looks a bit intimidating at first, but don't worry, we'll break it down step by step. Our mission is to take $-3(y+2)^2-5+6y$ and simplify it into its standard form. So, let's roll up our sleeves and get started!

Understanding the Problem

Before we jump into the simplification process, let's understand what we're dealing with. The expression is $-3(y+2)^2-5+6y$. The key part here is the $(y+2)^2$, which indicates we're going to be working with a quadratic expression. Our goal is to expand and combine like terms until we get the expression in the standard form, which is $ay^2 + by + c$, where a, b, and c are constants. To ace this, we need to remember the order of operations (PEMDAS/BODMAS) and how to expand squared binomials. This involves multiplying $(y + 2)$ by itself, distributing the -3, and then combining any similar terms we find along the way. This expression combines polynomial operations and simplification which is a core concept in algebra, often encountered when solving equations, graphing functions, and dealing with real-world problems involving quadratic relationships. Getting this right is crucial for many higher-level math concepts, including calculus and beyond. So, let's get it done!

Initial Expression Breakdown

Let's break down the expression $-3(y+2)^2-5+6y$ into its components to see what we are up against. We have a squared binomial, a constant term, and a linear term. Recognizing these parts helps us strategize our simplification process. The squared binomial $(y+2)^2$ tells us we'll need to expand this using the FOIL method (First, Outer, Inner, Last) or the binomial theorem. The constant term -5 is straightforward; we'll just need to combine it with any other constant terms we get after expanding. The linear term $6y$ will combine with any other 'y' terms we obtain during the expansion and distribution. This initial assessment allows us to organize our thoughts and tackle the problem systematically, ensuring no steps are missed and everything is accounted for in the final simplified form. By identifying these components, we are setting the stage for a smooth and accurate simplification process. This preliminary step is key to turning a complex-looking expression into something manageable and solvable. Now that we've broken it down, we can confidently move on to the expansion phase.

Step-by-Step Simplification

Okay, let's dive into the nitty-gritty of simplifying this expression. We'll take it one step at a time to make sure we don't miss anything. Remember, the goal is to transform $-3(y+2)^2-5+6y$ into the standard form of a quadratic expression: $ay^2 + by + c$.

1. Expand the Squared Binomial

First up, we need to tackle $(y+2)^2$. This means we're multiplying $(y+2)$ by itself. We can use the FOIL method (First, Outer, Inner, Last) or simply remember the pattern: $(a+b)^2 = a^2 + 2ab + b^2$. Let's apply this:

$(y+2)^2 = (y+2)(y+2)$

$= y*y + y*2 + 2*y + 2*2$

$= y^2 + 2y + 2y + 4$

$= y^2 + 4y + 4$

So, $(y+2)^2$ expands to $y^2 + 4y + 4$. This expansion is a critical first step. It transforms the expression from a factored form into a polynomial that we can further manipulate. Understanding how to correctly expand binomial squares is fundamental in algebra, and it's a skill you'll use repeatedly. Whether you choose to use the FOIL method or the binomial square formula, the key is to ensure each term is multiplied correctly. This foundational step sets the stage for the rest of the simplification, so getting it right is crucial. With this expansion done, we're ready to move on to the next stage of our simplification journey, where we'll distribute the -3 across the expanded trinomial.

2. Distribute the -3

Now that we've expanded $(y+2)^2$ to $y^2 + 4y + 4$, we need to distribute the -3 across these terms. This means multiplying each term inside the parenthesis by -3:

$-3(y^2 + 4y + 4) = -3*y^2 + (-3)*4y + (-3)*4$

$= -3y^2 - 12y - 12$

So, distributing the -3 gives us $-3y^2 - 12y - 12$. This distribution step is crucial because it removes the parentheses, allowing us to combine like terms later. It's important to pay close attention to the signs here, as a mistake with the negative sign can throw off the entire calculation. Each term inside the parentheses must be multiplied by -3, and the signs of the resulting terms need to be adjusted accordingly. This step demonstrates the distributive property of multiplication over addition and subtraction, a fundamental concept in algebra. Once we've distributed the -3 correctly, we can move on to the next step, where we'll bring in the remaining terms from the original expression and prepare to combine everything into a simplified form. So far, so good!

3. Rewrite the Expression

Let's bring everything together now. Our original expression was $-3(y+2)^2 - 5 + 6y$. We've expanded and distributed, so we can rewrite the expression with our new terms:

$-3y^2 - 12y - 12 - 5 + 6y$

This step is simply about gathering all the pieces we've worked on so far. We're taking the result of our distribution, $-3y^2 - 12y - 12$, and adding back the remaining terms from the original expression, which are -5 and +6y. Rewriting the expression in this way helps us see all the terms together in one place, making it easier to identify and combine like terms in the next step. It’s a crucial step for organization, ensuring that we don’t accidentally leave out any terms during the simplification process. The goal here is clarity, setting us up for accurate calculations as we proceed. With all the terms laid out before us, we're now in a good position to simplify the expression to its final form. Let's move on to combining those like terms!

4. Combine Like Terms

Now comes the fun part – combining like terms! We look for terms with the same variable and exponent. In our expression, $-3y^2 - 12y - 12 - 5 + 6y$, we have:

• $y^2$ terms: Only $-3y^2$
• $y$ terms: $-12y$ and $+6y$
• Constants: $-12$ and $-5$

Let's combine the y terms and the constants:

• $-12y + 6y = -6y$
• $-12 - 5 = -17$

So, combining like terms gives us $-3y^2 - 6y - 17$. This step is at the heart of simplifying algebraic expressions. It involves identifying terms that can be grouped together because they share the same variable and exponent, and then adding or subtracting their coefficients. For the $y$ terms, we combined -12y and 6y, resulting in -6y. For the constants, we combined -12 and -5, giving us -17. The $y^2$ term, -3y^2, remains unchanged because there are no other $y^2$ terms to combine it with. This process of combining like terms reduces the complexity of the expression, making it easier to understand and work with. By performing this step accurately, we're one step closer to the standard form and the final answer. We've now simplified the expression as much as possible, and it's time to present it in the correct format.

5. Write in Standard Form

The standard form of a quadratic expression is $ay^2 + by + c$. Looking at our simplified expression, $-3y^2 - 6y - 17$, we can see that it's already in the standard form:

• $a = -3$
• $b = -6$
• $c = -17$

So, our final simplified expression in standard form is $-3y^2 - 6y - 17$. Writing the expression in standard form is the final touch that brings clarity and order to our result. Standard form makes it easy to identify the coefficients and the constant term, which is essential for further analysis or problem-solving. In this case, our simplified expression, $-3y^2 - 6y - 17$, perfectly fits the $ay^2 + by + c$ format. This means we have successfully transformed the original expression into its simplest and most organized form. Recognizing and writing expressions in standard form is a fundamental skill in algebra, particularly when dealing with quadratic equations and functions. It's the standard way mathematicians and students communicate these expressions, ensuring everyone is on the same page. With this final step completed, we can confidently present our answer and know it’s in the correct and universally understood format.

Final Answer

After simplifying the expression $-3(y+2)^2-5+6y$, we have arrived at the standard form:

$-3y^2 - 6y - 17$

This is our final answer! We've successfully expanded, distributed, combined like terms, and arranged the expression in standard form. Great job, guys! You've taken a seemingly complex expression and simplified it step by step. Remember, breaking down the problem into smaller, manageable steps is key to tackling these kinds of challenges. By following this method, you can simplify any quadratic expression and confidently arrive at the correct answer. This process not only helps in simplifying expressions but also enhances your understanding of algebraic principles, setting you up for success in more advanced math topics. Keep practicing, and you'll become a pro at simplifying expressions in no time! Each problem you solve adds to your skills and confidence. So, let's keep learning and growing together! You've got this!