Evaluate $x^2+9x-8$ At $x=5$: A Step-by-Step Guide

by Luna Greco 55 views

Hey guys! Today, we're diving into the exciting world of algebraic expressions. Specifically, we're going to tackle the expression $x^2 + 9x - 8$ and figure out its value when $x$ is equal to 5. Sounds like fun, right? Trust me, it is! This is a fundamental skill in mathematics, and mastering it will set you up for success in more advanced topics. We'll break it down step-by-step, so even if algebra feels a bit intimidating right now, you'll be a pro by the end of this guide.

Understanding the Expression

Before we jump into plugging in numbers, let's make sure we understand what the expression $x^2 + 9x - 8$ actually means. At its heart, this is a combination of terms involving the variable $x$. The variable, in this case, $x$, represents an unknown value, and in this particular scenario, we're told that $x = 5$. The expression consists of three terms: $x^2$, $9x$, and $-8$. Let's dissect each of them:

  • $x^2$: This term represents $x$ raised to the power of 2, which is simply $x$ multiplied by itself ($x \times x$). When we substitute $x = 5$, this becomes $5^2$, or $5 \times 5$, which equals 25. Remember the order of operations! Exponents come before multiplication and addition.
  • $9x$: This term signifies 9 multiplied by $x$. The number 9 is what we call a coefficient – it's the numerical factor that multiplies the variable. So, when $x = 5$, the term $9x$ becomes $9 \times 5$, which equals 45. It's crucial to recognize that the absence of an operation symbol between 9 and $x$ implies multiplication. This is a common convention in algebra, and understanding it will prevent errors.
  • $-8$: This is a constant term. It's a number that stands alone without any variables attached. Constant terms always remain the same, regardless of the value of $x$. So, in our case, the constant term is simply -8. Don't forget to include the negative sign! It's an integral part of the term.

By understanding each part of the expression, we've laid a solid foundation for the next step: substitution. Substitution is the process of replacing the variable $x$ with its given value, which in this case is 5. This is a crucial step in evaluating expressions, and it's essential to be precise and careful to avoid mistakes.

The Substitution Step: Plugging in $x = 5$

Alright, now for the fun part! We're going to substitute $x = 5$ into our expression $x^2 + 9x - 8$. This means we'll replace every instance of $x$ with the number 5. It's like we're swapping out the unknown for its known value. Let's see how it looks:

Original expression: $x^2 + 9x - 8$

Substitute $x = 5$: $(5)^2 + 9(5) - 8$

Notice how we've enclosed the 5 in parentheses? This is a good practice, especially when dealing with negative numbers or more complex expressions. It helps to keep things organized and prevents confusion with the order of operations. The parentheses clearly indicate that we're squaring 5 and multiplying 9 by 5.

Now that we've successfully substituted, our expression has transformed from an algebraic expression with a variable to a numerical expression that we can easily simplify. The next step involves following the order of operations, often remembered by the acronym PEMDAS (or BODMAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Mastering this order is crucial for accurate calculations in mathematics and is a cornerstone of algebraic manipulation.

Applying the Order of Operations (PEMDAS/BODMAS)

Now that we've substituted $x = 5$ into our expression, we have $(5)^2 + 9(5) - 8$. It's time to put our PEMDAS (or BODMAS) knowledge to the test! Remember, this handy acronym tells us the order in which we need to perform the operations:

  1. Parentheses/Brackets: While we have parentheses in our expression, they're mainly used for clarity after the substitution. There aren't any operations to perform inside the parentheses in this case, so we can move on to the next step.
  2. Exponents/Orders: Ah, here we have an exponent! We need to evaluate $(5)^2$, which means 5 raised to the power of 2. As we discussed earlier, this is simply $5 \times 5$, which equals 25. So, we can replace $(5)^2$ with 25 in our expression.

Our expression now looks like this: $25 + 9(5) - 8$

  1. Multiplication and Division: Next up, we have multiplication. We see $9(5)$, which means 9 multiplied by 5. This gives us 45. There's no division in our expression, so we're good to go after handling this multiplication.

Our expression now looks like this: $25 + 45 - 8$

  1. Addition and Subtraction: Finally, we're left with addition and subtraction. We perform these operations from left to right. First, we add 25 and 45, which gives us 70. Then, we subtract 8 from 70, which leaves us with 62.

So, after carefully following the order of operations, we've arrived at our final answer: 62. It's like solving a puzzle, where each step builds upon the previous one, leading us to the ultimate solution. Understanding and applying PEMDAS is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving, a skill that's valuable in many areas of life.

The Final Result: The Expression Evaluated

Phew! We've made it through the substitution, navigated the order of operations, and arrived at our final answer. After all the steps, we can confidently say that when $x = 5$, the expression $x^2 + 9x - 8$ evaluates to 62. That's it! We've successfully evaluated the expression. It's like reaching the summit of a mathematical mountain – a sense of accomplishment and a clear view of the landscape.

In summary, here's a recap of the steps we took:

  1. Understanding the expression: We broke down the expression into its individual terms and understood what each term represented.
  2. Substitution: We replaced the variable $x$ with its given value, 5.
  3. Order of operations (PEMDAS/BODMAS): We carefully followed the order of operations to simplify the expression, ensuring we performed the operations in the correct sequence.
  4. Calculation: We performed the arithmetic operations to arrive at the final numerical value.

This process might seem like a lot of steps at first, but with practice, it becomes second nature. The more you work with algebraic expressions, the more comfortable you'll become with the substitution process and the application of the order of operations. Think of it as building muscle memory for your mathematical brain! You'll start to see patterns, anticipate steps, and even develop your own shortcuts.

The result, 62, is a single numerical value that represents the expression's worth when $x$ is 5. It's a concrete answer, a definitive solution to our initial problem. This process of evaluating expressions is fundamental in algebra and serves as a building block for more complex mathematical concepts. It's used in solving equations, graphing functions, and even in real-world applications like calculating the trajectory of a projectile or determining the cost of materials for a project. So, the skills you've honed today are not just about solving textbook problems; they're about equipping yourself with tools that you can use in various situations.

Practice Makes Perfect: Further Exploration

Now that you've seen how to evaluate this expression, the best way to solidify your understanding is to practice! Try plugging in different values for $x$ and see how the expression changes. What happens if $x$ is negative? What if $x$ is zero? What if $x$ is a fraction? Exploring these scenarios will deepen your understanding of how the variable $x$ affects the value of the expression. Consider these as challenges – opportunities to test your skills and expand your knowledge.

Here are a few suggestions for further exploration:

  • Try different values for $x$: Substitute $x = -2$, $x = 0$, $x = 1/2$, and see what results you get. Pay close attention to how negative signs and fractions affect the calculations.
  • Modify the expression: Change the coefficients (the numbers multiplying $x$) or the constant term. For example, try evaluating $2x^2 - 3x + 1$ when $x = 3$. This will help you understand how different components of the expression contribute to its overall value.
  • Create your own expressions: Challenge yourself to create your own algebraic expressions and evaluate them for different values of $x$. This is a great way to test your understanding and develop your problem-solving skills.

Evaluating expressions is a fundamental skill in algebra, and the more you practice, the better you'll become. Remember, math isn't just about memorizing formulas; it's about understanding concepts and developing a logical approach to problem-solving. So, keep exploring, keep practicing, and keep having fun with math! You've got this!

  • Evaluate the expression when $x=5$.