Evaluate $r^2 + (q+q)/6$ For Q=-15 And R=-3

Hey guys! Today, we're diving into a fun math problem where we need to evaluate an algebraic expression. It might look a bit intimidating at first, but trust me, we'll break it down step by step and it'll all make sense. We're given the expression $r^2 + (q+q)/6$, and we know the values of $q$ and $r$. Specifically, $q = -15$ and $r = -3$. So, our mission is to plug these values into the expression and simplify it to get our final answer. Think of it like a puzzle – we have all the pieces, we just need to put them in the right places! This kind of problem is super important in algebra because it helps us understand how variables and expressions work. It’s like building the foundation for more complex math later on. So, let's roll up our sleeves and get started! We'll go through each step slowly and carefully, so you can follow along and really understand what's happening. By the end of this, you'll be a pro at evaluating expressions like this. Remember, math is like a game – it's all about understanding the rules and applying them correctly. And with a little practice, you can master any math challenge that comes your way. Let's jump in and solve this expression together!

Step-by-Step Solution

Okay, let's get started with the solution. The first thing we need to do is substitute the given values of $q$ and $r$ into our expression. Remember, our expression is $r^2 + (q+q)/6$, and we have $q = -15$ and $r = -3$. So, wherever we see a $q$ in the expression, we're going to replace it with $-15$, and wherever we see an $r$, we'll replace it with $-3$. This is a crucial first step because it sets up the rest of the problem. If we don't substitute correctly, our final answer won't be right. So, pay close attention here! After we substitute, our expression will look something like this: $(-3)^2 + (-15 + -15)/6$. Notice how we've put the $-3$ and $-15$ in parentheses. This is important because it helps us keep track of the negative signs and ensures that we perform the operations in the correct order. Now that we've substituted the values, we're ready to move on to the next step, which involves simplifying the expression using the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? We'll be using that a lot in the next few steps, so make sure you're comfortable with it. Keep following along, and we'll get to the final answer in no time!

1. Substitute the Values

Alright, let's dive deeper into the first crucial step: substituting the values. As we mentioned earlier, we need to replace the variables $r$ and $q$ in the expression $r^2 + (q+q)/6$ with their given values, which are $r = -3$ and $q = -15$. This might seem straightforward, but it's super important to be careful with the negative signs. Trust me, a small mistake here can throw off the entire calculation. So, take your time and double-check your work. When we substitute $r$ with $-3$, we get $(-3)^2$. Notice the parentheses around the $-3$. This is absolutely essential because it tells us that we're squaring the entire number $-3$, not just the 3. If we didn't use parentheses, we might mistakenly calculate $-3^2$ as $-(3^2)$, which is $-9$, instead of $(-3)^2$, which is $9$. Big difference, right? Next, let's substitute $q$ with $-15$ in the fraction part of the expression, which is $(q+q)/6$. This becomes $(-15 + -15)/6$. Again, the parentheses are our friends here. They help us keep the negative signs straight and ensure we add the numbers correctly. So, after the substitution, our expression looks like this: $(-3)^2 + (-15 + -15)/6$. We've successfully replaced the variables with their values, and we're one step closer to solving the problem. Now, we're ready to move on to the next step, which is simplifying the expression using the order of operations. Remember PEMDAS? It's our guiding principle here.

2. Simplify the Exponent

Okay, now that we've substituted the values, it's time to simplify the expression. Remember PEMDAS? The first thing we need to tackle is the exponent. In our expression, $(-3)^2 + (-15 + -15)/6$, we have $(-3)^2$. This means we need to multiply $-3$ by itself. Now, here's a key thing to remember: a negative number multiplied by a negative number gives us a positive number. So, $(-3) * (-3) = 9$. This is a fundamental rule in math, and it's crucial for getting the correct answer here. If we made a mistake and thought $(-3)^2$ was $-9$, the whole problem would go off track. So, always double-check your signs! By simplifying the exponent, we've transformed $(-3)^2$ into $9$. Our expression now looks like this: $9 + (-15 + -15)/6$. See how we're slowly chipping away at the complexity of the expression? We've taken care of the exponent, and now we can move on to the next step in PEMDAS, which is dealing with the parentheses. We have a sum inside the parentheses that we need to simplify before we can do anything else. So, let's get to it! We're making great progress, and we're getting closer and closer to the final answer. Keep up the awesome work!

3. Simplify Inside the Parentheses

Alright, let's tackle what's inside the parentheses. Looking at our expression, $9 + (-15 + -15)/6$, we see that we have $(-15 + -15)$ inside the parentheses. Remember, adding a negative number is the same as subtracting the positive version of that number. So, $-15 + -15$ is the same as $-15 - 15$. Think of it like owing someone 15 dollars and then owing them another 15 dollars. How much do you owe in total? You owe 30 dollars, which we represent as $-30$. So, $(-15 + -15) = -30$. We've successfully simplified the expression inside the parentheses! Our expression now looks like this: $9 + (-30)/6$. We've made some serious progress. We've handled the exponent, simplified the parentheses, and now we're ready to move on to the next operation in PEMDAS, which is division. We have a fraction, $(-30)/6$, that we need to simplify. Remember, a fraction is just another way of writing a division problem. So, let's divide $-30$ by $6$ and see what we get. We're on the home stretch now, guys! Just a few more steps and we'll have our final answer. Keep pushing, and let's conquer this math problem together!

4. Perform the Division

Okay, it's time to tackle the division. Our expression currently looks like this: $9 + (-30)/6$. We have the fraction $(-30)/6$, which represents $-30$ divided by $6$. When we divide a negative number by a positive number, the result is a negative number. So, we know our answer will be negative. Now, let's think about the division itself. How many times does 6 go into 30? It goes in 5 times. Therefore, $-30$ divided by $6$ is $-5$. We've successfully performed the division! Our expression now looks much simpler: $9 + (-5)$. See how we're breaking down the problem into smaller, more manageable steps? That's the key to tackling any math challenge. By focusing on one operation at a time, we can avoid getting overwhelmed and ensure we get the correct answer. Now that we've done the division, we're left with a simple addition problem. We need to add $9$ and $-5$. This is the final step, guys! We're so close to the finish line. Let's add these numbers together and get our final answer. You've come this far, so let's make sure we nail it!

5. Perform the Addition

Alright, we've reached the final step! Our expression is now $9 + (-5)$. This is a simple addition problem, but it's important to remember how to add positive and negative numbers. Adding a negative number is the same as subtracting the positive version of that number. So, $9 + (-5)$ is the same as $9 - 5$. Think of it like having 9 dollars and then spending 5 dollars. How much money do you have left? You have 4 dollars. So, $9 - 5 = 4$. Therefore, $9 + (-5) = 4$. We've done it! We've successfully simplified the entire expression. We started with $r^2 + (q+q)/6$, substituted $q = -15$ and $r = -3$, and step-by-step, we simplified it down to our final answer. This was a journey, but we made it through together! Now, let's state our final answer clearly so that everyone knows what we've accomplished. You've shown some serious math skills today, guys. Give yourselves a pat on the back! Understanding how to evaluate expressions like this is a fundamental skill in algebra, and you've mastered it. Let's write down our final answer and celebrate!

Final Answer

So, after all our hard work, we've arrived at the final answer! We started with the expression $r^2 + (q+q)/6$, plugged in the values $q = -15$ and $r = -3$, and went through each step of simplifying using the order of operations (PEMDAS). We tackled the exponent, simplified the parentheses, performed the division, and finally, completed the addition. And what did we get? Drumroll, please... The final answer is $4$. That's right, guys! $r^2 + (q+q)/6 = 4$ when $q = -15$ and $r = -3$. You did an amazing job following along and working through this problem. Remember, math is like a muscle – the more you use it, the stronger it gets. So, keep practicing, keep challenging yourself, and keep exploring the wonderful world of math. You've proven that you have what it takes to solve complex problems, and I'm super proud of you. Now, go forth and conquer more mathematical challenges! You've got this!