Simplify -3(y+2)^2-5+6y: A Step-by-Step Guide

Hey guys! Today, we're diving deep into the world of algebraic expressions, specifically tackling the simplification of a seemingly complex expression: $-3(y+2)^2-5+6y$. Don't worry, it's not as daunting as it looks! We'll break it down step-by-step, making sure you understand the why behind each move. So, grab your pencils and let's get started!

Deciphering the Expression: Our Starting Point

The expression we're working with is $-3(y+2)^2-5+6y$. At first glance, it might seem like a jumble of numbers, variables, and parentheses. But, like any good puzzle, there's a clear path to solving it. Our goal is to simplify this expression, which means we want to rewrite it in a more concise and understandable form, typically in standard form. This involves expanding, combining like terms, and arranging the terms in descending order of their exponents. This process makes the expression easier to work with and interpret. Think of it as decluttering your room – you're organizing the elements to make everything more accessible and functional. Understanding the order of operations is key here. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is our roadmap for simplifying the expression correctly. We'll use this order to guide our steps and ensure we arrive at the right answer. So, before we jump into the calculations, let's take a moment to appreciate the structure of the expression and the journey we're about to embark on. By understanding the individual components and the order in which we need to address them, we set ourselves up for success in simplifying this algebraic puzzle.

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

The first hurdle we need to clear is expanding the squared term, $(y+2)^2$. This isn't just a matter of squaring each term inside the parentheses; we need to remember the rules of expanding binomials. When we square a binomial like $(y+2)$, we're essentially multiplying it by itself: $(y+2)(y+2)$. There are a couple of ways to approach this. One is the FOIL method (First, Outer, Inner, Last), which helps us remember to multiply each term in the first binomial by each term in the second. The other is to use the binomial square formula: $(a+b)^2 = a^2 + 2ab + b^2$. Both methods will lead us to the same result. Let's use the FOIL method for this example. First, we multiply the First terms: $y * y = y^2$. Then, the Outer terms: $y * 2 = 2y$. Next, the Inner terms: $2 * y = 2y$. And finally, the Last terms: $2 * 2 = 4$. Now, we add these together: $y^2 + 2y + 2y + 4$. We can combine the like terms (the two $2y$ terms) to get $y^2 + 4y + 4$. So, $(y+2)^2$ expands to $y^2 + 4y + 4$. This is a crucial step in simplifying the expression, as it allows us to get rid of the parentheses and work with individual terms. We've now transformed a potentially confusing part of the expression into a more manageable form. Remember, expanding binomials is a fundamental skill in algebra, and mastering it will make simplifying expressions much easier.

Step 2: Distributing the -3: Multiplying Through

Now that we've expanded $(y+2)^2$ to $y^2 + 4y + 4$, we need to deal with the $-3$ that's multiplying it. This is where the distributive property comes into play. The distributive property states that $a(b+c) = ab + ac$. In other words, we need to multiply the $-3$ by each term inside the parentheses. So, we have $-3(y^2 + 4y + 4)$. Multiplying $-3$ by $y^2$ gives us $-3y^2$. Multiplying $-3$ by $4y$ gives us $-12y$. And multiplying $-3$ by $4$ gives us $-12$. Therefore, $-3(y^2 + 4y + 4)$ becomes $-3y^2 - 12y - 12$. It's important to pay close attention to the signs here. A negative multiplied by a positive results in a negative, and a negative multiplied by a negative results in a positive. This step is crucial because it removes the parentheses and allows us to combine like terms in the next step. We're essentially distributing the multiplication across the entire expression within the parentheses, ensuring that each term is correctly accounted for. By carefully applying the distributive property, we transform a product into a sum of terms, making the expression easier to manipulate and simplify further. This step highlights the power of algebraic properties in simplifying complex expressions.

Step 3: Combining Like Terms: Bringing it Together

With the expansion and distribution done, our expression now looks like this: $-3y^2 - 12y - 12 - 5 + 6y$. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms with $y$ to the power of 1 ($-12y$ and $6y$) and two constant terms ($-12$ and $-5$). To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). Let's start with the $y$ terms. We have $-12y + 6y$. Adding the coefficients, we get $-12 + 6 = -6$. So, $-12y + 6y$ simplifies to $-6y$. Next, let's combine the constant terms. We have $-12 - 5$. Subtracting 5 from -12 gives us $-17$. Now, we can rewrite the expression with the combined like terms: $-3y^2 - 6y - 17$. Notice that the $-3y^2$ term remains unchanged because there are no other terms with $y^2$ to combine it with. Combining like terms is a fundamental step in simplifying algebraic expressions. It allows us to reduce the number of terms and make the expression more concise. By identifying and combining like terms, we're essentially tidying up the expression and making it easier to understand and work with.

Step 4: Standard Form: The Final Touch

We've done the heavy lifting – expanding, distributing, and combining like terms. Now, the final step is to write the simplified expression in standard form. Standard form for a quadratic expression (an expression with a term raised to the power of 2) is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, our expression is $-3y^2 - 6y - 17$. Comparing this to the standard form, we can see that it's already in the correct order! The term with the highest power of the variable ($y^2$) comes first, followed by the term with the next highest power ($y$), and finally the constant term. So, our simplified expression in standard form is $-3y^2 - 6y - 17$. Writing an expression in standard form is important because it makes it easier to compare and analyze different expressions. It also helps with further algebraic manipulations, such as factoring or solving equations. By arranging the terms in a consistent order, we create a clear and organized representation of the expression. This final step ensures that our simplified expression is not only mathematically correct but also presented in a way that is easy to understand and use.

The Grand Finale: Our Simplified Product

We've successfully navigated the world of algebraic simplification! By carefully expanding, distributing, combining like terms, and arranging in standard form, we've transformed the original expression, $-3(y+2)^2-5+6y$, into its simplified form: $-3y^2 - 6y - 17$. Woohoo! This journey demonstrates the power of algebraic techniques in making complex expressions more manageable. Each step we took was crucial in unraveling the expression and revealing its underlying structure. Remember, simplifying expressions is a fundamental skill in algebra, and it's essential for solving equations, graphing functions, and tackling more advanced mathematical concepts. So, pat yourselves on the back for mastering this skill! You're now well-equipped to take on more algebraic challenges. And remember, practice makes perfect. The more you work with simplifying expressions, the more confident and proficient you'll become.

So, the simplified product in standard form is:

oxed{-3}y^2 + \boxed{-6}y + \boxed{-17}

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