Which Product Is Equivalent To $25x^2 - 16$?A. $(5x - 4)(5x + 4)$ B. \$(5x + 8)(5x - 8)$[/tex\] C. $(5x - 4)(5x - 4)$ D. $(5x - 8)(5x - 8)$
**Which Product is Equivalent to $25x^2 - 16$?**
Understanding the Problem
In this article, we will explore the concept of equivalent products in algebra and help you determine which product is equivalent to the given expression $25x^2 - 16$. We will break down the problem step by step and provide a clear explanation of the solution.
What are Equivalent Products?
Equivalent products are expressions that have the same value when simplified. In other words, they are expressions that can be transformed into each other through algebraic operations. Understanding equivalent products is crucial in algebra as it helps us simplify complex expressions and solve equations.
Step 1: Factorize the Given Expression
To determine which product is equivalent to $25x^2 - 16$, we need to factorize the given expression. Factorization involves breaking down an expression into simpler components. In this case, we can factorize $25x^2 - 16$ as follows:
Using the Difference of Squares Formula
We can use the difference of squares formula to simplify the expression further:
Applying this formula to our expression, we get:
Comparing the Factored Expression with the Options
Now that we have factored the given expression, we can compare it with the options provided:
A. $(5x - 4)(5x + 4)$ B. $(5x + 8)(5x - 8)$ C. $(5x - 4)(5x - 4)$ D. $(5x - 8)(5x - 8)$
Which Product is Equivalent to $25x^2 - 16$?
Based on our factorization, we can see that the correct answer is:
A. $(5x - 4)(5x + 4)$
This is because our factored expression matches option A exactly.
Conclusion
In this article, we have explored the concept of equivalent products in algebra and helped you determine which product is equivalent to the given expression $25x^2 - 16$. We have broken down the problem step by step and provided a clear explanation of the solution. By understanding equivalent products, you can simplify complex expressions and solve equations with ease.
Frequently Asked Questions
Q: What is the difference of squares formula?
A: The difference of squares formula is $(a^2 - b^2) = (a + b)(a - b)$.
Q: How do I factorize an expression?
A: To factorize an expression, you need to break it down into simpler components. You can use the difference of squares formula to simplify expressions of the form $(a^2 - b^2)$.
Q: What is the significance of equivalent products in algebra?
A: Equivalent products are crucial in algebra as they help you simplify complex expressions and solve equations. By understanding equivalent products, you can transform complex expressions into simpler ones, making it easier to solve equations.
Q: How do I determine which product is equivalent to a given expression?
A: To determine which product is equivalent to a given expression, you need to factorize the expression and compare it with the options provided. You can use the difference of squares formula to simplify expressions of the form $(a^2 - b^2)$.
Q: What is the correct answer to the problem?
A: The correct answer to the problem is A. $(5x - 4)(5x + 4)$.
Q: Can I use the difference of squares formula to simplify other types of expressions?
A: Yes, you can use the difference of squares formula to simplify expressions of the form $(a^2 - b^2)$. However, you need to be careful when applying the formula to ensure that the expression can be factored in the same way.
Q: How do I apply the difference of squares formula?
A: To apply the difference of squares formula, you need to identify the values of and in the expression $(a^2 - b^2)$. You can then use the formula to simplify the expression.
Q: What are some common mistakes to avoid when working with equivalent products?
A: Some common mistakes to avoid when working with equivalent products include:
- Not factoring the expression correctly
- Not using the difference of squares formula when applicable
- Not comparing the factored expression with the options provided
- Not being careful when applying the difference of squares formula
By avoiding these common mistakes, you can ensure that you are working with equivalent products correctly and accurately.