Completing Algebraic Tables: Step-by-Step Solutions

by Aria Freeman 52 views

Hey guys! Today, we are diving into some awesome algebraic exercises. We're going to break down how to complete a table by performing different procedures and finding the results. This is super useful for understanding how expressions work and how to simplify them. Let's jump right in and make math fun!

Understanding the Basics

Before we get started, let’s quickly recap some fundamental concepts. When we talk about algebraic expressions, we're dealing with variables (like x) and constants (numbers) combined with operations (like addition, subtraction, multiplication, and division). The goal is often to simplify these expressions or to find the value of the variable that makes the expression true.

Key Concepts to Remember

  • Variables: Letters that represent unknown values (e.g., x, y, z).
  • Constants: Numbers that have a fixed value (e.g., 3, 9, 4).
  • Terms: Parts of an expression separated by addition or subtraction.
  • Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and 5x are like terms, but 3x and 3x² are not).
  • Distributive Property: a(b + c) = ab + ac (this is super important for expanding expressions).
  • Combining Like Terms: Adding or subtracting like terms to simplify expressions.

Exercise 1: x(x + 3) = x² + 3x

Let’s kick things off with our first example: x(x + 3) = x² + 3x. In this exercise, we're applying the distributive property. The distributive property, guys, is like the superhero power of algebra! It lets us multiply a term outside the parentheses with each term inside the parentheses. So, what’s happening here is:

  • We start with x(x + 3).
  • We distribute the x across both terms inside the parentheses: x * x and x * 3.
  • This gives us x² + 3x.

Breaking It Down

  1. Distribute x to x: When you multiply x by x, you get x². Remember, x² means x times x.
  2. Distribute x to 3: When you multiply x by 3, you get 3x. It's just like having 3 groups of x.
  3. Combine the Results: Now, you add the results together: x² + 3x.

Why is this important? Understanding the distributive property is crucial for simplifying expressions and solving equations. It's like having a key that unlocks many doors in the world of algebra.

Exercise 2: 3(x + 3) = 3x + 9

Next up, we have 3(x + 3) = 3x + 9. This one is another awesome example of the distributive property in action! We're multiplying the constant 3 with each term inside the parentheses.

The Procedure Unveiled

  1. Start with 3(x + 3): We’ve got 3 multiplied by the expression (x + 3).
  2. Distribute 3 to x: When you multiply 3 by x, you get 3x. Imagine you have x, and you have three of them.
  3. Distribute 3 to 3: Multiply 3 by 3, which gives you 9. This is straightforward multiplication.
  4. Combine the Results: Add the results together: 3x + 9.

So, 3(x + 3) becomes 3x + 9. Pretty neat, huh?

Why This Matters

This exercise highlights how constants interact with variables inside expressions. It's a foundational concept for simplifying more complex algebraic expressions. Think of it as building a strong base for a mathematical skyscraper!

Exercise 3: (x + 3)(x + 3) = x² + 6x + 9

Now, let’s tackle something a bit more challenging: (x + 3)(x + 3) = x² + 6x + 9. Guys, this is where we start squaring binomials! A binomial is an expression with two terms, and squaring it means multiplying it by itself. This is also often written as (x + 3)².

Breaking Down the Multiplication

  1. Start with (x + 3)(x + 3): We are multiplying the binomial (x + 3) by itself.
  2. Use the FOIL Method (or Distribution): The FOIL method is a handy way to remember how to multiply two binomials. It stands for:
    • First: Multiply the first terms in each binomial (x * x = x²).
    • Outer: Multiply the outer terms (x * 3 = 3x).
    • Inner: Multiply the inner terms (3 * x = 3x).
    • Last: Multiply the last terms (3 * 3 = 9).
  3. Combine the Results: Add all the terms together: x² + 3x + 3x + 9.
  4. Combine Like Terms: Notice that we have two 3x terms. Add them together: 3x + 3x = 6x.

So, the final result is x² + 6x + 9.

The Significance of Squaring Binomials

Squaring binomials is a common operation in algebra and calculus. Understanding how to do this efficiently can save you a lot of time and reduce errors. Plus, it’s a stepping stone to understanding more advanced concepts like factoring and quadratic equations.

Completing the Table: The Next Steps

We've completed the first three exercises! Now, let's move on to the remaining exercises to complete the table. We have:

  • (x + 4)(x + 4) = ?
  • (x + 5)(x + 5) = ?

We’ll use the same principles we’ve learned so far to solve these. Remember, the key is to use the distributive property (or the FOIL method) and then combine like terms.

Exercise 4: (x + 4)(x + 4) = ?

Let’s break down (x + 4)(x + 4) step by step. This is very similar to the previous exercise, so we'll use the same approach.

The Procedure

  1. Start with (x + 4)(x + 4): We’re squaring the binomial (x + 4).
  2. Use the FOIL Method:
    • First: x * x = x²
    • Outer: x * 4 = 4x
    • Inner: 4 * x = 4x
    • Last: 4 * 4 = 16
  3. Combine the Results: Add all the terms together: x² + 4x + 4x + 16.
  4. Combine Like Terms: Add the 4x terms: 4x + 4x = 8x.

So, the final result is x² + 8x + 16.

Why This Matters

Practicing these exercises helps solidify your understanding of binomial multiplication. You’ll start to see patterns and become quicker at these types of problems. It's like building muscle memory for math!

Exercise 5: (x + 5)(x + 5) = ?

Last but not least, let’s tackle (x + 5)(x + 5). By now, you guys are probably getting the hang of this!

The Steps to Success

  1. Start with (x + 5)(x + 5): Squaring the binomial (x + 5).
  2. Apply the FOIL Method:
    • First: x * x = x²
    • Outer: x * 5 = 5x
    • Inner: 5 * x = 5x
    • Last: 5 * 5 = 25
  3. Combine the Results: Add all the terms together: x² + 5x + 5x + 25.
  4. Combine Like Terms: Add the 5x terms: 5x + 5x = 10x.

Therefore, the final result is x² + 10x + 25.

The Big Picture

Completing this exercise gives you a solid foundation in algebraic manipulation. You've now seen how to expand and simplify expressions involving binomials, which is a critical skill in algebra.

Summing It Up

Okay, guys, we’ve completed all the exercises! Let’s recap what we've done and why it's important.

Key Takeaways

  • Distributive Property: We used this to multiply terms outside parentheses with terms inside.
  • FOIL Method: This helped us multiply two binomials efficiently.
  • Combining Like Terms: Simplifying expressions by adding or subtracting terms with the same variable and power.
  • Squaring Binomials: A common operation in algebra that involves multiplying a binomial by itself.

By understanding these concepts, you're well-equipped to tackle more complex algebraic problems. Keep practicing, and you'll become a math whiz in no time!

The Completed Table

Here’s a summary of the results in a table format:

Exercise Procedure Result
x(x + 3) Distribute x x² + 3x
3(x + 3) Distribute 3 3x + 9
(x + 3)(x + 3) FOIL Method, Combine Like Terms x² + 6x + 9
(x + 4)(x + 4) FOIL Method, Combine Like Terms x² + 8x + 16
(x + 5)(x + 5) FOIL Method, Combine Like Terms x² + 10x + 25

Final Thoughts

Algebraic expressions might seem intimidating at first, but breaking them down step by step makes them much more manageable. Remember, practice makes perfect! Keep working on these skills, and you'll find that algebra becomes a powerful tool in your mathematical arsenal. You've got this!

Keep up the great work, and see you in the next math adventure!