Evaluate $p^2 + 13$ When $p = -4$: A Simple Guide

Hey guys! Today, we're diving into a super straightforward math problem. We're going to figure out what the value of the expression $p^2 + 13$ is when $p$ is equal to $-4$. Don't worry; it's much easier than it sounds! So, grab your thinking caps, and let's get started!

Understanding the Expression

Before we jump right in, let's break down what the expression $p^2 + 13$ actually means. In math terms, $p^2$ means $p$ multiplied by itself (i.e., $p * p$). The "plus 13" part just means we're adding 13 to whatever we get when we square $p$. Essentially, we need to substitute the value of p into the expression and perform the calculation. This type of problem is fundamental in algebra and understanding how to evaluate expressions is a crucial skill. Whether you're a student just starting out with algebra or someone looking to brush up on your math skills, this guide will help you understand each step of the process. We'll start by plugging in the value of $p$, then we'll perform the necessary arithmetic operations to arrive at the final answer. Make sure you follow along and practice similar problems to build your confidence. Remember, math can be fun and rewarding, especially when you understand the basics. Keep practicing and you'll be surprised at how quickly you improve! Also, keep in mind the order of operations (PEMDAS/BODMAS) which dictates that exponents should be handled before addition. This is a key concept to ensure we get the right answer. With that said, let's go ahead and begin solving our expression and unraveling this mathematical puzzle together, one step at a time!

Substituting the Value of $p$

Okay, so the first thing we need to do is substitute the value of $p$, which is $-4$, into our expression. So, wherever we see $p$ in the expression $p^2 + 13$, we're going to replace it with $-4$. This gives us $(-4)^2 + 13$. Remember to put $-4$ in parentheses. This is super important because it tells us that we're squaring the entire negative number. If we didn't use parentheses, we might accidentally think we're only squaring the 4 and then making it negative, which would give us the wrong answer. Pay close attention to signs in mathematics. Now, let's pause and recap why this substitution step is critical. In algebra, we often work with variables, which are symbols that represent unknown values. In our case, $p$ is a variable, and we're given its specific value: $-4$. Substituting this value allows us to transform the expression from an algebraic one with a variable into a numerical one that we can actually calculate. Without substitution, we'd be stuck with an expression we can't simplify. The substitution process turns the expression from abstract to concrete, allowing us to find a numerical result. Think of it like replacing a placeholder with its actual value in a sentence; it gives the sentence meaning. In mathematical terms, it allows us to evaluate the expression and find its worth when $p$ is $-4$. It's a foundational skill used throughout mathematics and is crucial for solving equations and understanding more complex mathematical concepts. Therefore, always double-check that you've substituted the value correctly and that you've maintained the integrity of the expression by using parentheses when necessary.

Calculating $(-4)^2$

Next up, we need to figure out what $(-4)^2$ means. Remember, anything squared means that number multiplied by itself. So, $(-4)^2$ is the same as $(-4) * (-4)$. Now, here's a quick refresher on multiplying negative numbers: a negative number multiplied by a negative number gives you a positive number. So, $(-4) * (-4)$ equals $16$. So, $(-4)^2$ is $16$. This is a crucial point to understand because getting the sign wrong can throw off your entire calculation. The concept of squaring a negative number coming out positive is used frequently in various branches of math including geometry and calculus. Now that we know that $(-4)^2 = 16$, we can move on to the next step, which is adding 13 to this result. This will give us the final answer to our original question. But before we proceed, let's delve a bit deeper into why a negative times a negative is a positive. Think of it like owing someone money. If you have a debt (negative), and that debt is taken away (another negative), you're essentially gaining money (positive). Mathematically, this concept can be proven using the properties of real numbers and the distributive property. Understanding the underlying logic can help you remember the rule and apply it correctly in different situations. So, whether you're dealing with simple arithmetic or complex algebraic equations, remembering that a negative times a negative is a positive will be a valuable asset in your mathematical toolkit. Now, armed with this knowledge and understanding, let's confidently proceed to the final step of our problem and discover the final solution.

Adding 13

Now that we know $(-4)^2 = 16$, we can substitute that back into our expression. This gives us $16 + 13$. To find the final answer, we simply add 16 and 13 together. What is $16 + 13$? It's $29$! And that's it! We've solved the problem. Therefore, the final answer is $29$! Just to recap, we first substituted $-4$ for $p$ in the expression $p^2 + 13$, giving us $(-4)^2 + 13$. Then, we calculated $(-4)^2$, which is $16$. Finally, we added $16$ and $13$ to get $29$. Therefore, the value of $p^2 + 13$ when $p = -4$ is $29$. Woohoo! We've done it! This final step underscores the importance of basic addition skills in mathematics. While the problem itself involves algebraic concepts, the ultimate calculation relies on the ability to accurately add numbers together. A solid foundation in arithmetic is essential for success in more advanced mathematical topics. In this case, adding 16 and 13 might seem straightforward, but it's crucial to double-check your work to avoid careless errors. Even the smallest mistake can lead to an incorrect final answer. Also, remember that addition is commutative, which means that the order in which you add the numbers doesn't matter. So, $16 + 13$ is the same as $13 + 16$. This property can sometimes be helpful when performing calculations in your head or when checking your work. With that said, congratulations on reaching the end of this step and successfully completing the problem. You've demonstrated your understanding of substitution, exponents, and addition, which are all valuable skills in mathematics!

Final Answer

So, to wrap it up, if $p = -4$, then $p^2 + 13 = 29$. Great job guys! Understanding how to evaluate expressions like this is a fundamental skill in algebra, and you've nailed it! Keep practicing, and you'll become a math whiz in no time! Remember, math is like building blocks; each concept builds upon the previous one. So, mastering the basics, like evaluating expressions, is essential for tackling more complex problems in the future. Whether you're solving for variables in equations, graphing functions, or working with geometric shapes, the skills you've learned here will come in handy. Always remember to follow the order of operations (PEMDAS/BODMAS), pay attention to signs, and double-check your work to avoid errors. And most importantly, don't be afraid to ask for help when you need it. Math can be challenging, but it's also rewarding. The more you practice and persevere, the better you'll become. So, keep up the great work, and never stop learning! With a little effort and dedication, you can achieve anything you set your mind to. Now that we've successfully evaluated the expression $p^2 + 13$ when $p = -4$, you're well-equipped to tackle similar problems with confidence and ease. Keep practicing and exploring new mathematical concepts, and you'll be amazed at how far you can go!