Which Expression Is Equivalent to 10x²y + 25x²?

Understanding how to simplify algebraic expressions is a crucial skill in mathematics. which expression is equivalent to 10x2y + 25×2? In this article, we will focus on the question: “Which expression is equivalent to 10x²y + 25x²?” We will explore the process of simplifying this expression and identifying its equivalent forms by breaking down the steps involved. Mastery of these techniques provides a deeper understanding of algebraic manipulation and problem-solving.

Breaking Down the Expression

To determine which expression is equivalent to 10x²y + 25x², we first need to analyze its components. The given expression consists of two terms:

10x²y: This term includes a coefficient (10), a variable raised to the second power (x²), and another variable (y).
25x²: This term has a coefficient (25) and a variable raised to the second power (x²).

Since both terms share the common factor x², we can use this factor to simplify the expression. Identifying the greatest common factor (GCF) is crucial for the next steps.

Factoring the Common Factor

To find which expression is equivalent to 10x²y + 25x², we factor out the GCF. Here, the GCF is 5x². Factoring 5x² from each term, we get:

$10 x^{2} y + 25 x^{2} = 5 x^{2} (2 y) + 5 x^{2} (5)$

This simplifies to:

$5 x^{2} (2 y + 5)$

Thus, 5x²(2y + 5) is the equivalent expression to 10x²y + 25x².

Verifying the Equivalent Expression

To confirm that 5x²(2y + 5) is truly equivalent to 10x²y + 25x², we expand it back:

$\cdot 2y + 5x² \cdot 5 = 10x²y + 25x²$

This expansion verifies that 5x²(2y + 5) matches the original expression, proving that it is an equivalent expression to 10x²y + 25x².

Understanding the Structure of Equivalent Expressions

When asked which expression is equivalent to 10x²y + 25x², recognizing the structure of the simplified form 5x²(2y + 5) helps in understanding how algebraic expressions can be manipulated. This factored form clearly shows the common factor 5x² distributed over the terms 2y and 5. This technique simplifies complex expressions and facilitates easier analysis and manipulation in algebraic contexts.

Application of Equivalent Expressions

Knowing which expression is equivalent to 10x²y + 25x² is beneficial in various mathematical applications. Simplified expressions, such as 5x²(2y + 5), make it easier to solve equations and perform algebraic operations. This skill is also vital in calculus, where simplification is often necessary before differentiation or integration.

Expanding the Concept: Factoring Complex Expressions

The technique used to determine which expression is equivalent to 10x²y + 25x² can be applied to more complex algebraic expressions. For instance, consider the expression 20x³y² + 35x²y. The GCF here is 5x²y, and factoring it out yields:

$20 x^{3} y^{2} + 35 x^{2} y = 5 x^{2} y (4 x y + 7)$

This process of identifying the GCF and simplifying can be applied to a wide range of expressions, demonstrating its universal applicability in algebra.

The Importance of Practice

To effectively determine which expression is equivalent to 10x²y + 25x², practice is key. Working through various problems involving different expressions helps in recognizing patterns and common factors more quickly. Like any mathematical skill, frequent practice makes the process more intuitive and enhances problem-solving abilities.

Conclusion

In summary, understanding which expression is equivalent to 10x²y + 25x² involves simplifying the expression by factoring out the greatest common factor. which expression is equivalent to 10x2y + 25×2?
The equivalent expression 5x²(2y + 5) not only matches the original expression but also facilitates easier manipulation and analysis. Mastering these techniques is essential for solving algebraic equations and performing advanced mathematical operations, underscoring the importance of algebraic simplification in mathematics. Read More D2armorpicker.