Simplify (7x^2+3)(7x^2-3) Using Difference Of Squares

Jul 18, 2025 by ADMIN 54 views

Simplify (7x^2+3)(7x^2-3) Using the Difference of Squares Formula

Hey guys! Let's dive into simplifying the expression extbf{ $(7x^2 + 3)(7x^2 - 3)$ } using the difference of squares formula. This is a classic algebraic problem, and understanding how to tackle it can really boost your math skills. The difference of squares formula is a super handy tool that makes simplifying expressions like this a breeze. So, grab your pencils, and let's get started!

Understanding the Difference of Squares Formula

Before we jump into the problem, let's quickly recap the extit{difference of squares formula}. It states that for any two terms, 'a' and 'b', the product of their sum and difference is equal to the difference of their squares. Mathematically, it looks like this:

(a + b)(a - b) = a^2 - b^2

This formula is a shortcut that helps us avoid the lengthy process of expanding the product using the distributive property (also known as FOIL – First, Outer, Inner, Last). Recognizing this pattern can save you time and effort in many algebraic problems. Now, let’s understand why this formula works. Imagine you're multiplying $(a + b)$ by $(a - b)$ . If you use the distributive property, you'd multiply 'a' by both 'a' and '-b', and then 'b' by both 'a' and '-b'. This gives you $a^2 - ab + ba - b^2$ . Notice that the middle terms, $-ab$ and $+ba$ , cancel each other out because they are additive inverses. This leaves us with the simplified form $a^2 - b^2$ . This understanding is crucial because it allows us to directly apply the formula without having to go through the entire expansion process each time. It’s like having a superpower for algebra! In our case, we can see how this formula applies beautifully. We have two terms, $7x^2$ and $3$ , which are being added and subtracted, perfectly fitting the $(a + b)(a - b)$ structure. Recognizing this pattern is the first step in efficiently simplifying the expression, and it highlights the elegance and efficiency of algebraic formulas.

Applying the Formula to (7x^2+3)(7x2-3)

Now, let's apply the extbf{difference of squares formula} to our expression: $(7x^2 + 3)(7x^2 - 3)$ . If we compare this to the formula $(a + b)(a - b) = a^2 - b^2$ , we can see that:

$a = 7x^2$
$b = 3$

So, all we need to do is substitute these values into the formula. This means we'll square 'a' (which is $7x^2$ ) and subtract the square of 'b' (which is 3). Let's break it down step-by-step. First, we find $a^2$ , which is $(7x^2)^2$ . Remember that when you raise a product to a power, you raise each factor to that power. So, $(7x^2)^2$ becomes $7^2 * (x^2)^2$ . We know that $7^2$ is 49. For $(x^2)^2$ , we use the power of a power rule, which states that $(x^m)^n = x^{m*n}$ . In this case, it means we multiply the exponents 2 and 2, giving us $x^4$ . Therefore, $(7x^2)^2$ simplifies to $49x^4$ . Next, we need to find $b^2$ , which is $3^2$ . This is simply 3 multiplied by itself, which equals 9. Now, we have all the pieces we need. According to the difference of squares formula, $(7x^2 + 3)(7x^2 - 3) = a^2 - b^2$ . Substituting the values we found, we get $49x^4 - 9$ . This is the simplified form of the original expression. Isn't it neat how the formula allows us to skip the more tedious multiplication process and jump straight to the answer? By recognizing the pattern and applying the formula correctly, we've efficiently simplified a seemingly complex expression. This technique is a cornerstone of algebra and will be useful in many more advanced problems you encounter.

Step-by-Step Solution

Let's go through the extit{solution step-by-step} to make sure we've got it all clear:

Identify the pattern: Recognize that $(7x^2 + 3)(7x^2 - 3)$ fits the form $(a + b)(a - b)$ .
Determine 'a' and 'b':
- $a = 7x^2$
- $b = 3$
Apply the formula: $(a + b)(a - b) = a^2 - b^2$
Substitute 'a' and 'b': $(7x^2 + 3)(7x^2 - 3) = (7x^2)^2 - (3)^2$
Simplify:
- $(7x^2)^2 = 49x^4$
- $(3)^2 = 9$
Final result: $49x^4 - 9$

Each of these steps is crucial to arriving at the correct answer. First, recognizing the pattern is like unlocking a secret code – it tells you which tool (in this case, the difference of squares formula) to use. Identifying 'a' and 'b' is simply a matter of matching the terms in your expression to the variables in the formula. Applying the formula is the heart of the process, where you replace the original product with the difference of the squares. The substitution step is where you plug in the values you identified for 'a' and 'b'. Finally, simplifying involves performing the necessary mathematical operations, such as squaring terms and dealing with exponents. In our example, squaring $7x^2$ requires understanding how to apply the exponent to both the coefficient (7) and the variable term ( $x^2$ ). Similarly, squaring 3 is straightforward. The final step brings it all together, giving you the simplified expression. This step-by-step approach not only leads to the correct solution but also builds a strong foundation for tackling more complex algebraic problems. By breaking down the process into manageable steps, you can improve your accuracy and confidence in algebra.

The Correct Answer

So, after applying the difference of squares formula and simplifying, we found that:

(7x^2 + 3)(7x^2 - 3) = 49x^4 - 9

Therefore, the correct answer is D) $49x^4 - 9$ . It's awesome how a simple formula can make a big difference, right? Using the difference of squares formula not only simplifies the expression but also does so in a very elegant and efficient way. This is the power of mathematical tools – they allow us to approach problems with a structured and strategic mindset. Options A, B, and C are incorrect because they don't fully account for the squaring of both the coefficient and the variable term, or they miss the correct application of the difference of squares pattern. For example, option A, $7x^2 - 1$ , completely misses the squaring aspect of both terms. Option B, $7x^4 - 9$ , correctly squares the variable term but fails to square the coefficient 7. Option C, $49x^2 - 9$ , correctly squares the coefficient but incorrectly keeps the variable term as $x^2$ instead of $x^4$ . Understanding why these options are incorrect reinforces the importance of each step in the simplification process, from correctly identifying 'a' and 'b' to applying the formula and simplifying the result. This careful attention to detail is what leads to success in algebra and mathematics in general. By understanding the common mistakes and how to avoid them, you're not just getting the right answer – you're also building a deeper understanding of the underlying principles.

Tips for Mastering the Difference of Squares

To really nail the extbf{difference of squares}, here are a few tips that can help you:

Practice makes perfect: The more you practice, the easier it will become to recognize the pattern. Try working through a variety of examples.
Memorize the formula: Knowing the formula $(a + b)(a - b) = a^2 - b^2$ by heart is crucial. It's your go-to tool for these types of problems.
Pay attention to detail: Make sure you're squaring both the coefficients and the variables correctly. A small mistake can lead to a wrong answer.
Check your work: After simplifying, take a moment to review your steps and ensure you haven't made any errors.
Understand the 'why': Don't just memorize the formula; understand why it works. This will help you apply it in different situations.

These tips are your roadmap to mastering the difference of squares. Practice, as in any skill, is essential. The more you work through examples, the faster you'll recognize the pattern and the more confident you'll become in applying the formula. Memorizing the formula is like having a key that unlocks the solution – it's the fundamental tool you need. However, memorization should be coupled with understanding. Knowing why the formula works gives you a deeper appreciation for the mathematics and allows you to apply the concept in various contexts, not just in textbook problems. Paying attention to detail is where accuracy comes in. Algebra involves many small steps, and a mistake in any one of them can throw off the entire solution. This is especially true when squaring terms; you need to make sure you're applying the exponent to every factor within the term. Checking your work is the final safeguard. It's a chance to catch any errors you might have made and ensure that your answer is correct. By making it a habit to review your steps, you can significantly improve your accuracy. Remember, mastering the difference of squares isn't just about getting the right answer to one specific type of problem; it's about building a solid foundation in algebra that will serve you well in more advanced topics. So, keep practicing, stay focused on the details, and embrace the elegance and power of mathematical formulas!

Conclusion

So there you have it! Simplifying $(7x^2 + 3)(7x^2 - 3)$ using the extbf{difference of squares formula} is a straightforward process once you understand the formula and how to apply it. Remember to look for the pattern, identify 'a' and 'b', and then substitute them into the formula. With a little practice, you'll be a pro at this in no time! Keep up the great work, and happy simplifying! This formula is not just a trick; it's a fundamental algebraic tool that you'll encounter again and again in your mathematical journey. Mastering it now will make many future problems easier to tackle. Think of it as an investment in your mathematical skills. The more comfortable you are with the difference of squares, the more efficiently you can solve problems and the more you can focus on the core concepts rather than getting bogged down in the algebra. Moreover, the ability to recognize patterns in mathematics is a crucial skill that extends far beyond this specific formula. It's a way of thinking that will help you in various areas of math and even in other fields. So, by mastering the difference of squares, you're not just learning a formula; you're honing your pattern-recognition skills and developing a more sophisticated approach to problem-solving. This kind of understanding and skill-building is what makes learning mathematics so rewarding. You're not just memorizing facts; you're learning how to think critically and solve problems effectively. So, keep exploring, keep practicing, and keep challenging yourself. The world of mathematics is full of fascinating patterns and tools, and the more you learn, the more you'll appreciate the beauty and power of this subject.