Solving The Algebraic Expression (x+4a)*(5a-8x)-(6+7x)*(3x+2a)

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Solving the Algebraic Expression (x+4a)*(5a-8x)-(6+7x)*(3x+2a)

Hey guys! Let's dive into solving this algebraic expression step by step. Algebraic expressions can seem daunting at first, but breaking them down makes it much easier. So, if you've stumbled upon this article, you're in the right place to learn how to tackle such problems. We're going to take a conversational and friendly approach, ensuring you not only understand the solution but also the logic behind it.

Understanding the Problem

Before we jump into the solution, let's quickly recap what an algebraic expression is. Think of it as a mathematical phrase that can contain numbers, variables (like our x and a), and operations (addition, subtraction, multiplication, division, etc.). In our case, the expression we need to solve is (x+4a)\(5a-8x)-(6+7x)\(3x+2a). It involves multiplying binomials and then subtracting the results. To solve this, we'll use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last) and combine like terms.

The first part of the expression, (x + 4a) * (5a - 8x), involves multiplying two binomials. Remember the distributive property? It's our best friend here! This property tells us that each term in the first binomial must be multiplied by each term in the second binomial. Let's break it down:

  1. First: Multiply the first terms in each binomial: x * 5a = 5ax.
  2. Outer: Multiply the outer terms: x * -8x = -8x^2.
  3. Inner: Multiply the inner terms: 4a * 5a = 20a^2.
  4. Last: Multiply the last terms: 4a * -8x = -32ax.

So, when we put it all together, we get 5ax - 8x^2 + 20a^2 - 32ax. Now, let's simplify this by combining like terms. We have two terms with ax: 5ax and -32ax. Combining these gives us -27ax. So, the simplified form of the first part is -8x^2 - 27ax + 20a^2.

The second part of the expression, (6 + 7x) * (3x + 2a), is another set of binomials to multiply. We'll use the same distributive property technique here. Let's go through it step by step:

  1. First: Multiply the first terms: 6 * 3x = 18x.
  2. Outer: Multiply the outer terms: 6 * 2a = 12a.
  3. Inner: Multiply the inner terms: 7x * 3x = 21x^2.
  4. Last: Multiply the last terms: 7x * 2a = 14ax.

Combining these, we get 18x + 12a + 21x^2 + 14ax. This part doesn’t have any like terms to combine, so we'll keep it as is for now.

Step-by-Step Solution

Now, let's break down the solution into manageable steps.

Step 1: Expand the First Part of the Expression

We'll start by expanding the first part: (x + 4a) * (5a - 8x). To do this, we'll use the distributive property (FOIL method) which we discussed earlier. This means each term in the first parenthesis will multiply each term in the second parenthesis.

  • x * 5a = 5ax
  • x * -8x = -8x^2
  • 4a * 5a = 20a^2
  • 4a * -8x = -32ax

So, expanding (x + 4a) * (5a - 8x) gives us 5ax - 8x^2 + 20a^2 - 32ax.

Step 2: Simplify the First Part

Next, we combine like terms in the expanded form. We have two terms with ax: 5ax and -32ax. Adding these together, we get -27ax. So, the simplified form of the first part is:

-8x^2 - 27ax + 20a^2

This makes our expression much cleaner and easier to work with.

Step 3: Expand the Second Part of the Expression

Now, let's tackle the second part: (6 + 7x) * (3x + 2a). Again, we'll use the distributive property (FOIL method) to expand this.

  • 6 * 3x = 18x
  • 6 * 2a = 12a
  • 7x * 3x = 21x^2
  • 7x * 2a = 14ax

So, expanding (6 + 7x) * (3x + 2a) gives us 18x + 12a + 21x^2 + 14ax.

Step 4: Simplify the Second Part

There are no like terms to combine in this part, so we leave it as is:

18x + 12a + 21x^2 + 14ax

This part is already in a simplified form, ready to be used in the next step.

Step 5: Combine and Simplify the Entire Expression

Now, we'll combine the simplified forms of both parts. Remember, the original expression was (x + 4a) * (5a - 8x) - (6 + 7x) * (3x + 2a). We've simplified the first part to -8x^2 - 27ax + 20a^2 and the second part to 18x + 12a + 21x^2 + 14ax. Now, we subtract the second part from the first:

(-8x^2 - 27ax + 20a^2) - (18x + 12a + 21x^2 + 14ax)

To subtract, we distribute the negative sign across the second expression:

-8x^2 - 27ax + 20a^2 - 18x - 12a - 21x^2 - 14ax

Now, let's combine like terms:

  • x^2 terms: -8x^2 - 21x^2 = -29x^2
  • ax terms: -27ax - 14ax = -41ax
  • a^2 terms: 20a^2 (no other a^2 terms)
  • x terms: -18x (no other x terms)
  • a terms: -12a (no other a terms)

So, the final simplified expression is:

-29x^2 - 41ax + 20a^2 - 18x - 12a

Final Answer

Therefore, the simplified form of the given algebraic expression (x+4a)(5a-8x)-(6+7x)(3x+2a)** is:

-29x^2 - 41ax + 20a^2 - 18x - 12a

Key Takeaways

  • Distributive Property (FOIL): This is crucial for expanding binomials. Remember to multiply each term in the first parenthesis by each term in the second parenthesis.
  • Combining Like Terms: After expanding, always look for like terms to simplify the expression. This makes the expression cleaner and easier to manage.
  • Step-by-Step Approach: Complex expressions become manageable when you break them down into smaller steps. Expand, simplify, and then combine.

Common Mistakes to Avoid

  • Sign Errors: Be extra careful when distributing negative signs. A common mistake is forgetting to distribute the negative to all terms inside the parenthesis.
  • Incorrect Multiplication: Double-check your multiplication, especially when dealing with variables and coefficients.
  • Missing Terms: Ensure you multiply each term correctly. It's easy to miss a term, especially in longer expressions.

Practice Makes Perfect

The best way to get comfortable with algebraic expressions is practice! Try solving similar problems on your own. You can also look for online resources and practice worksheets. Remember, algebra is like a puzzle – the more you practice, the better you become at solving it!

Conclusion

And there you have it, guys! We've successfully solved the algebraic expression (x+4a)(5a-8x)-(6+7x)(3x+2a)**. Remember, the key to solving these problems is to break them down into manageable steps and take your time. Don't rush, and always double-check your work. Keep practicing, and you'll become an algebra whiz in no time! If you have any questions or want to try another example, feel free to drop a comment below. Happy solving!