Factor The Expression $72x^2 - 50y^2$.

Jun 18, 2025 by ADMIN 39 views

Factoring $72x^2 - 50y^2$ - A Comprehensive Guide

Factoring expressions is a fundamental skill in algebra, and mastering it opens doors to solving complex equations and simplifying mathematical problems. In this comprehensive guide, we will delve into the process of factoring the expression $72x^2 - 50y^2$ . This seemingly simple expression holds within it a wealth of mathematical concepts and techniques that, once understood, will significantly enhance your algebraic prowess. We will explore the various steps involved, from identifying common factors to applying the difference of squares pattern. By the end of this guide, you will not only be able to factor this specific expression but also possess a deeper understanding of factoring in general. Let's embark on this mathematical journey together and unlock the secrets hidden within $72x^2 - 50y^2$ .

1. Identifying Common Factors

The first step in factoring any algebraic expression is to identify common factors among the terms. In the expression $72x^2 - 50y^2$ , we observe two terms: $72x^2$ and $50y^2$ . Our goal is to find the greatest common factor (GCF) that divides both terms evenly. The GCF is the largest number or expression that can be factored out of all the terms.

To find the GCF of 72 and 50, we can list their factors:

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Factors of 50: 1, 2, 5, 10, 25, 50

The common factors are 1 and 2, with 2 being the greatest common factor. Now, let's consider the variable parts of the terms. The first term, $72x^2$ , has $x^2$ , and the second term, $50y^2$ , has $y^2$ . Since there are no common variables between the terms, the GCF for the variables is 1. Therefore, the GCF of the entire expression is 2.

Factoring out the GCF, we rewrite the expression as:

$72x^2 - 50y^2 = 2(36x^2 - 25y^2)$

This step simplifies the expression and sets the stage for further factoring. We have successfully identified and factored out the common factor of 2, leaving us with a new expression inside the parentheses that we can now focus on.

2. Recognizing the Difference of Squares Pattern

After factoring out the common factor, we now have the expression $2(36x^2 - 25y^2)$ . Looking inside the parentheses, we can recognize a specific pattern: the difference of squares. The difference of squares pattern is a fundamental concept in algebra and is expressed as:

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

This pattern states that the difference between two perfect squares can be factored into the product of the sum and the difference of their square roots. To apply this pattern, we need to ensure that our expression fits the form $a^2 - b^2$ .

In our case, we have $36x^2 - 25y^2$ . We can rewrite this as:

$(6x)^2 - (5y)^2$

Here, $a = 6x$ and $b = 5y$ . We can see that $36x^2$ is the square of $6x$ , and $25y^2$ is the square of $5y$ . This confirms that our expression indeed fits the difference of squares pattern. Recognizing this pattern is crucial because it allows us to apply a specific factoring technique that simplifies the expression significantly.

3. Applying the Difference of Squares Formula

Having identified the difference of squares pattern in the expression $36x^2 - 25y^2$ , we can now apply the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$ . We've already established that $a = 6x$ and $b = 5y$ . Substituting these values into the formula, we get:

$(6x)^2 - (5y)^2 = (6x + 5y)(6x - 5y)$

This step directly applies the pattern we recognized, transforming the difference of two squares into a product of two binomials. The expression $(6x + 5y)(6x - 5y)$ represents the factored form of $36x^2 - 25y^2$ . It's a significant simplification, breaking down a complex expression into more manageable components. This factorization is a key step in solving equations, simplifying fractions, and other algebraic manipulations.

4. Completing the Factoring Process

We've successfully factored the expression inside the parentheses, but we must not forget the common factor we initially extracted. We found that $72x^2 - 50y^2 = 2(36x^2 - 25y^2)$ . Now that we have factored $36x^2 - 25y^2$ as $(6x + 5y)(6x - 5y)$ , we need to combine this result with the initial common factor.

So, the complete factoring of the original expression is:

$72x^2 - 50y^2 = 2(6x + 5y)(6x - 5y)$

This is the final factored form of the expression. We have broken down the original expression into its simplest factors: a constant (2) and two binomials $(6x + 5y)$ and $(6x - 5y)$ . This complete factorization provides valuable insights into the structure of the expression and is a critical step in various mathematical applications.

5. Verifying the Factored Form

To ensure the accuracy of our factoring, it's always a good practice to verify the factored form. We can do this by expanding the factored expression and checking if it matches the original expression. Let's expand our factored form, $2(6x + 5y)(6x - 5y)$ :

First, we expand the binomials $(6x + 5y)(6x - 5y)$ using the distributive property (also known as the FOIL method):

$(6x + 5y)(6x - 5y) = (6x)(6x) + (6x)(-5y) + (5y)(6x) + (5y)(-5y)$

$= 36x^2 - 30xy + 30xy - 25y^2$

Notice that the middle terms, $-30xy$ and $+30xy$ , cancel each other out:

$= 36x^2 - 25y^2$

Now, we multiply the result by the common factor 2:

$2(36x^2 - 25y^2) = 2(36x^2) - 2(25y^2)$

$= 72x^2 - 50y^2$

The expanded form, $72x^2 - 50y^2$ , matches our original expression. This confirms that our factoring is correct. Verification is a crucial step in the factoring process as it ensures that we haven't made any errors and that our factored form is equivalent to the original expression.

6. Practical Applications of Factoring

Factoring is not just an abstract mathematical exercise; it has numerous practical applications in various fields. Understanding how to factor expressions like $72x^2 - 50y^2$ can be incredibly useful in:

Solving Equations: Factoring is a key technique for solving quadratic equations and other polynomial equations. By factoring an equation, we can often find its roots or solutions more easily.
Simplifying Algebraic Fractions: Factoring both the numerator and the denominator of an algebraic fraction allows us to cancel out common factors, simplifying the fraction.
Calculus: Factoring is used in calculus to simplify expressions before differentiation or integration.
Engineering and Physics: Many problems in engineering and physics involve algebraic equations that need to be solved, and factoring is often a crucial step in the solution process.
Computer Science: Factoring techniques are used in computer science for optimizing algorithms and simplifying complex expressions.

By mastering factoring, you equip yourself with a powerful tool that can be applied in a wide range of real-world scenarios. The ability to break down complex expressions into simpler components is a valuable skill that enhances problem-solving abilities across various disciplines.

7. Conclusion

In conclusion, we have successfully factored the expression $72x^2 - 50y^2$ by following a step-by-step approach. We began by identifying and factoring out the greatest common factor, which was 2. This simplified the expression to $2(36x^2 - 25y^2)$ . We then recognized the difference of squares pattern within the parentheses, which allowed us to apply the formula $a^2 - b^2 = (a + b)(a - b)$ .

Applying this formula, we factored $36x^2 - 25y^2$ into $(6x + 5y)(6x - 5y)$ . Finally, we combined the common factor with the factored form to obtain the complete factorization: $72x^2 - 50y^2 = 2(6x + 5y)(6x - 5y)$ . We also verified our result by expanding the factored form and confirming that it matched the original expression.

This process not only demonstrates the specific factoring of $72x^2 - 50y^2$ but also illustrates the general principles of factoring algebraic expressions. Factoring is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts and applications. By understanding the techniques and patterns involved, you can tackle a wide range of factoring problems with confidence. The skills you've gained here will be invaluable in your further mathematical studies and in various real-world applications where algebraic manipulation is required.