Option A is equivalent to the given expression

Oct 13, 2025 by windailys.com 84 views

**Which Expressions are Equivalent to $$25 x^4 - 64$$?**

Understanding the Problem

In this article, we will explore the concept of equivalent expressions in algebra. Equivalent expressions are mathematical expressions that have the same value, but may be written in different forms. We will examine five different options and determine which ones are equivalent to the given expression $$25 x^4 - 64$$.

Option A: $$25 x^4 + 40 x - 40 x - 64$$

At first glance, this option may seem like a viable solution. However, upon closer inspection, we can see that the expression contains two terms that cancel each other out: $$40 x - 40 x$$. This means that the expression simplifies to $$25 x^4 - 64$$, which is the original expression. Therefore, Option A is equivalent to the given expression.

Option B: $$25 x^4 + 13 x - 13 x - 64$$

Similar to Option A, this expression also contains two terms that cancel each other out: $$13 x - 13 x$$. However, the presence of the $$+ 13 x$$ term means that this expression is not equivalent to the original expression. Therefore, Option B is not equivalent to the given expression.

Option C: $$(5 x^2 + 8)(5 x^2 - 8)$$

To determine if this expression is equivalent to the original expression, we need to expand the product using the distributive property. This gives us:

(5 x^2 + 8)(5 x^2 - 8) = 25 x^4 - 40 x^2 + 40 x^2 - 64

As we can see, the $$40 x^2$$ terms cancel each other out, leaving us with $$25 x^4 - 64$$, which is the original expression. Therefore, Option C is equivalent to the given expression.

Option D: $$(x^2 + 13)(x^2 - 13)$$

Using the distributive property, we can expand this product as follows:

(x^2 + 13)(x^2 - 13) = x^4 - 13 x^2 + 13 x^2 - 169

As we can see, the $$13 x^2$$ terms cancel each other out, leaving us with $$x^4 - 169$$. This expression is not equivalent to the original expression, which is $$25 x^4 - 64$$. Therefore, Option D is not equivalent to the given expression.

Conclusion

In conclusion, we have examined five different options and determined which ones are equivalent to the given expression $$25 x^4 - 64$$. The options that are equivalent to the given expression are:

Option A: $$25 x^4 + 40 x - 40 x - 64$$
Option C: $$(5 x^2 + 8)(5 x^2 - 8)$$

These expressions have the same value as the original expression, but may be written in different forms. The options that are not equivalent to the given expression are:

Option B: $$25 x^4 + 13 x - 13 x - 64$$
Option D: $$(x^2 + 13)(x^2 - 13)$$

Frequently Asked Questions

Q: What is the difference between equivalent expressions and equivalent equations? A: Equivalent expressions are mathematical expressions that have the same value, but may be written in different forms. Equivalent equations, on the other hand, are equations that have the same solution set.
Q: How can I determine if two expressions are equivalent? A: To determine if two expressions are equivalent, you can use the distributive property to expand the product of the two expressions. If the resulting expression is the same as the original expression, then the two expressions are equivalent.
Q: What is the importance of equivalent expressions in algebra? A: Equivalent expressions are important in algebra because they allow us to simplify complex expressions and make them easier to work with. They also help us to identify patterns and relationships between different mathematical expressions.

Additional Resources

Algebraic Expressions: This article provides an overview of algebraic expressions and how to simplify them.
Distributive Property: This article explains the distributive property and how to use it to expand products.
Equivalent Equations: This article discusses equivalent equations and how to determine if two equations are equivalent.