Calculate (3a-4b)^2: A Step-by-Step Guide

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Hey guys! Ever stumbled upon an equation that looks like a jumbled mess of letters and numbers? Don't sweat it! We're going to break down a common type of algebraic expression: squaring a binomial. In this guide, we'll specifically tackle the expression (3a - 4b)^2. Trust me, it's not as scary as it looks. By the end, you'll be a pro at solving these kinds of problems. So, grab your pencils, and let's dive in!

Understanding the Basics: What's a Binomial?

Before we jump into the calculation, let's quickly recap what a binomial is. Simply put, a binomial is an algebraic expression with two terms. These terms are connected by either an addition or subtraction sign. Think of examples like (x + y), (2a - b), or, in our case, (3a - 4b). Recognizing binomials is the first step in conquering these types of problems. The key characteristic to remember is the presence of two distinct terms. This is important because the method we use to square a binomial is different from squaring a single term or a trinomial (an expression with three terms).

Why This Matters

You might be wondering, "Why do I even need to know this?" Well, binomials pop up everywhere in algebra and beyond. From solving quadratic equations to understanding geometric formulas, mastering binomials is crucial. Plus, it builds a strong foundation for more advanced math concepts. So, understanding the binomial and how it behaves when squared is a fundamental skill. It unlocks a whole new level of problem-solving abilities in mathematics.

The Formula: (a - b)^2 = a^2 - 2ab + b^2

Alright, now for the magic formula! This is the key to unlocking the solution for expressions like (3a - 4b)^2. The general formula for squaring a binomial (a - b) is: (a - b)^2 = a^2 - 2ab + b^2. Let’s break this down:

  • (a - b)^2 means (a - b) multiplied by itself: (a - b) * (a - b).
  • a^2 means 'a' squared, or 'a' multiplied by itself.
  • -2ab means -2 times 'a' times 'b'.
  • b^2 means 'b' squared, or 'b' multiplied by itself.

This formula is derived from the distributive property of multiplication. When you multiply (a - b) by (a - b), you end up with this exact pattern. Memorizing this formula will save you tons of time and effort in the long run.

Mastering the Formula: A Crucial Step

The formula might seem abstract now, but it's the foundation for solving these problems efficiently. Think of it as a recipe – you need the right ingredients (the formula) to bake a delicious cake (solve the equation). Don't just memorize it; understand why it works. This understanding will make it easier to apply in different situations. So take a moment to really grasp the formula: (a - b)^2 = a^2 - 2ab + b^2.

Applying the Formula to (3a - 4b)^2

Now comes the fun part – using the formula to solve our specific problem! We have (3a - 4b)^2. Let's identify our 'a' and 'b' in this case:

  • a = 3a
  • b = 4b

See? It's just about recognizing the pattern. Now, we'll substitute these values into our formula: (a - b)^2 = a^2 - 2ab + b^2.

Step-by-Step Substitution

This is where careful substitution is key. We're replacing the abstract 'a' and 'b' with our specific terms, '3a' and '4b'. Take your time and double-check your work at each step. A small mistake here can throw off the whole calculation. So, let's substitute and see what we get!

Step-by-Step Calculation

Let's plug in our values and break down the calculation step by step:

  1. (3a)^2: This means (3a) * (3a). Remember to square both the 3 and the 'a'. So, (3a)^2 = 9a^2.
  2. -2 * (3a) * (4b): Multiply the numbers first: -2 * 3 * 4 = -24. Then, multiply the variables: a * b = ab. So, -2 * (3a) * (4b) = -24ab.
  3. (4b)^2: This means (4b) * (4b). Again, square both the 4 and the 'b'. So, (4b)^2 = 16b^2.

Now, let's put it all together:

(3a - 4b)^2 = 9a^2 - 24ab + 16b^2

And there you have it! We've successfully calculated (3a - 4b)^2.

The Importance of Order of Operations

Notice how we tackled each term separately? This follows the order of operations (PEMDAS/BODMAS), which is crucial in algebra. Squaring and multiplication take precedence over addition and subtraction. Sticking to this order ensures we get the correct result. Always double-check that you are following the order of operations!

Common Mistakes to Avoid

When squaring binomials, there are a few common pitfalls to watch out for:

  • Forgetting the middle term (-2ab): This is the most frequent mistake. People often square the first and last terms but forget to include the -2ab term. Remember, it's a crucial part of the formula!
  • Incorrectly squaring terms: Make sure you square both the coefficient (the number) and the variable. For example, (3a)^2 is 9a^2, not 3a^2.
  • Mixing up signs: Pay close attention to the signs, especially when dealing with subtraction. A wrong sign can completely change the answer.

Pro Tip: Double-Check Your Work

After you've solved the problem, take a moment to double-check your work. One way to do this is to actually multiply (3a - 4b) * (3a - 4b) using the distributive property. If you get the same answer as using the formula, you know you're on the right track. This habit of double-checking will significantly reduce errors.

Practice Problems

Okay, guys, time to put your newfound skills to the test! Here are a few practice problems for you to try:

  1. (2x + 5y)^2
  2. (a - 3c)^2
  3. (4m + n)^2

Work through these problems using the formula we discussed. Don't just look at the solutions – actually try solving them yourself. The more you practice, the more confident you'll become.

Solutions (No Peeking!)

  1. 4x^2 + 20xy + 25y^2
  2. a^2 - 6ac + 9c^2
  3. 16m^2 + 8mn + n^2

How did you do? If you got them right, awesome! If not, don't worry. Go back and review the steps, identify where you went wrong, and try again. Learning from your mistakes is a key part of mastering any math concept.

Conclusion

So, there you have it! Squaring the binomial (3a - 4b)^2 is no longer a mystery. By understanding the formula (a - b)^2 = a^2 - 2ab + b^2 and practicing the steps, you can confidently tackle these types of problems. Remember to avoid common mistakes, double-check your work, and keep practicing. The more you practice, the easier it will become. Keep up the great work, and you'll be a math whiz in no time!

I hope this guide has been helpful. If you have any questions, feel free to ask! Now go out there and conquer those binomials!