Multiply Binomials: Simplify (7a-16)(3a-7) Easily

Hey guys! Let's dive into simplifying the expression $(7a - 16)(3a - 7)$. We're going to break it down step-by-step, making sure we get to the simplest form possible. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the question is asking. We have two binomials, $(7a - 16)$ and $(3a - 7)$, and we need to multiply them together. This means we'll be using the distributive property, which some of you might know as the FOIL method (First, Outer, Inner, Last). This method helps us ensure that each term in the first binomial is multiplied by each term in the second binomial.

• Keywords are crucial, especially when dealing with mathematical problems. So, remember we are focusing on multiplying binomials and simplifying expressions. Simplifying means combining like terms to get our final answer in its most compact form. We will simplify the answer completely. That's our ultimate goal here.
• Now, let’s think about the big picture for a second. Why do we even simplify expressions like this? Well, in many real-world scenarios, mathematical models involve complex equations. Simplifying these equations makes them easier to work with, analyze, and solve. For instance, in physics, you might encounter equations that describe the motion of objects, and simplifying them can help you predict where an object will be at a certain time. Similarly, in economics, simplifying equations can help you understand market trends and make informed decisions. So, the skills we’re practicing here aren’t just abstract math concepts; they have practical applications in various fields.
• Moreover, mastering these algebraic manipulations is essential for more advanced mathematical topics. When you move on to calculus, trigonometry, or even statistics, you’ll find yourself using these fundamental skills constantly. So, by getting a solid grasp on simplifying expressions now, you’re setting yourself up for success in your future math courses. Think of it as building a strong foundation for a skyscraper – the stronger the foundation, the taller and more impressive the building can be. So, let’s make sure our foundation is rock solid!

Step-by-Step Solution

Let's use the FOIL method to expand the expression $(7a - 16)(3a - 7)$. Remember, FOIL stands for:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

Let's apply this step-by-step:

1. First: Multiply the first terms: $(7a) * (3a) = 21a^2$
2. Outer: Multiply the outer terms: $(7a) * (-7) = -49a$
3. Inner: Multiply the inner terms: $(-16) * (3a) = -48a$
4. Last: Multiply the last terms: $(-16) * (-7) = 112$

Now, let's put it all together:

$21a^2 - 49a - 48a + 112$

Combining Like Terms

Our next step is to simplify the expression by combining like terms. In this case, the like terms are the '-49a' and '-48a' terms. These are like terms because they both contain the variable 'a' raised to the power of 1.

Let's combine them:

$-49a - 48a = -97a$

Now, we can rewrite our expression as:

$21a^2 - 97a + 112$

Final Simplified Answer

Okay, guys, we've done the hard work! We've expanded the binomials using the FOIL method and combined like terms. Now, let's write out our final, simplified answer.

The final simplified form of $(7a - 16)(3a - 7)$ is:

f{21a^2 - 97a + 112}

This expression cannot be simplified further because there are no more like terms to combine. We have a quadratic expression in its standard form, and we’re done!

• Checking our work is crucial to ensure we haven't made any mistakes along the way. A simple way to check is to substitute a value for 'a' into both the original expression and the simplified expression. If we get the same result, we can be confident that our simplification is correct. For example, let’s try substituting a = 1:

  • Original expression: $(7(1) - 16)(3(1) - 7) = (7 - 16)(3 - 7) = (-9)(-4) = 36$
  • Simplified expression: $21(1)^2 - 97(1) + 112 = 21 - 97 + 112 = 36$

  Since both expressions give us the same result (36), this gives us some confidence that our simplification is correct. However, it’s worth noting that this check isn’t foolproof – there’s still a small chance we could have made an error that this specific value doesn’t catch. For a more rigorous check, you could try substituting a different value for 'a', or even expand the simplified expression back to the original form to see if they match.

• Understanding the structure of the simplified expression is also important. The expression $21a^2 - 97a + 112$ is a quadratic expression, which means it has the form $ax^2 + bx + c$, where a, b, and c are constants. In this case, a = 21, b = -97, and c = 112. Quadratic expressions are encountered frequently in mathematics and physics, and understanding their properties is crucial. For example, the graph of a quadratic expression is a parabola, and the roots (or solutions) of the equation $21a^2 - 97a + 112 = 0$ represent the points where the parabola intersects the x-axis. Knowing these connections can be incredibly useful in solving problems and understanding the behavior of various systems.

Common Mistakes to Avoid

Hey, we all make mistakes, especially when we're learning something new! But being aware of common pitfalls can help us avoid them. When multiplying binomials and simplifying, there are a few typical errors that students often make.

1. Forgetting to distribute properly: This is a big one! Remember, each term in the first binomial needs to be multiplied by each term in the second binomial. If you miss a multiplication, your answer will be incorrect. That's why using the FOIL method can be helpful – it provides a structured way to ensure you don’t miss any terms.
2. Incorrectly multiplying signs: Watch out for those negative signs! A negative times a negative is a positive, and a negative times a positive is a negative. It's super easy to make a mistake here, so double-check your signs as you go. This is especially important when you have terms like (-16) * (-7), where a sign error can completely change the result.
3. Combining unlike terms: You can only combine terms that have the same variable raised to the same power. For example, you can combine '-49a' and '-48a' because they both have 'a' to the power of 1, but you can't combine '21a^2' with '-97a' because one has 'a^2' and the other has 'a'. Mixing up like and unlike terms is a common mistake that can lead to incorrect simplifications.
4. Simple arithmetic errors: Even if you understand the concept perfectly, a simple mistake in multiplication or addition can throw off your entire answer. This is why it's always a good idea to double-check your calculations, especially in the intermediate steps. Using a calculator for complex multiplications can also help reduce the risk of arithmetic errors.

To avoid these mistakes, practice is key. The more you work through these types of problems, the more comfortable you'll become with the process, and the less likely you'll be to make errors. Remember, math is like a sport – you get better with practice!

Practice Problems

Alright guys, now that we've gone through the solution and discussed common mistakes, it's time to put your knowledge to the test! Practice is crucial for mastering any mathematical concept, so let's work through a few more examples.

Here are a couple of problems for you to try. Remember to use the FOIL method, combine like terms, and simplify completely:

1. $(5x + 3)(2x - 1)$
2. $(4y - 6)(3y + 2)$

Work these out on your own, and then you can check your answers. The goal is not just to get the right answer, but also to understand the process of getting there. Think about each step you're taking, and why you're taking it. This will help you develop a deeper understanding of the underlying concepts.

Let's go through the solutions to these practice problems. This will give you a chance to check your work and see if you've made any mistakes. Even if you got the correct answer, it’s still helpful to see the step-by-step solution, as it can reinforce your understanding and highlight any alternative approaches.

Problem 1: $(5x + 3)(2x - 1)$

• First: $(5x)(2x) = 10x^2$
• Outer: $(5x)(-1) = -5x$
• Inner: $(3)(2x) = 6x$
• Last: $(3)(-1) = -3$

Combining these, we get:

$10x^2 - 5x + 6x - 3$

Now, combine like terms:

$-5x + 6x = x$

So, the simplified answer is:

f{10x^2 + x - 3}

Problem 2: $(4y - 6)(3y + 2)$

• First: $(4y)(3y) = 12y^2$
• Outer: $(4y)(2) = 8y$
• Inner: $(-6)(3y) = -18y$
• Last: $(-6)(2) = -12$

Combining these, we get:

$12y^2 + 8y - 18y - 12$

Now, combine like terms:

$8y - 18y = -10y$

So, the simplified answer is:

f{12y^2 - 10y - 12}

How did you do? Did you get the correct answers? If not, don't worry! Go back and review your steps, and see if you can identify where you went wrong. Mistakes are a part of the learning process, and they can be valuable opportunities for growth.

Conclusion

Awesome job, guys! We've successfully multiplied the binomials $(7a - 16)(3a - 7)$ and simplified the result completely. We used the FOIL method, combined like terms, and arrived at our final answer: $21a^2 - 97a + 112$. Remember to practice these steps, and you'll become a pro at simplifying expressions in no time!