Factor $25y^2 + 120y + 144$: A Step-by-Step Guide

Factoring quadratic expressions can sometimes feel like solving a puzzle, but with the right approach, it becomes a straightforward process. In this article, we'll break down the steps to factor the quadratic expression $25y^2 + 120y + 144$ completely and simplify it. Whether you're a student tackling algebra or just looking to refresh your math skills, this guide will provide a clear, step-by-step explanation.

Recognizing the Quadratic Expression

Before diving into the factoring process, it's essential to recognize the structure of the given expression. The expression $25y^2 + 120y + 144$ is a quadratic expression in the form of $ay^2 + by + c$, where:

• $a = 25$
• $b = 120$
• $c = 144$

This form is crucial because it dictates the methods we can use to factor the expression. One common approach is to look for two binomials that, when multiplied together, yield the original quadratic expression. This involves finding two numbers that multiply to $ac$ and add up to $b$. Understanding this basic structure sets the stage for efficient factoring. Now, let's delve into the process of factoring this specific quadratic expression, ensuring each step is clear and easy to follow. This initial recognition is the key to unlocking the solution and simplifying the expression effectively. Remember, the goal is to rewrite the quadratic expression as a product of two binomials, which will ultimately lead to a simpler, factored form.

Step-by-Step Factoring Process

To factor the quadratic expression $25y^2 + 120y + 144$, we'll follow a structured approach that makes the process manageable and clear. Here’s how we'll do it:

1. Check for a Common Factor

First, always check if there's a common factor that can be factored out from all terms. In this case, the terms $25y^2$, $120y$, and $144$ do not share any common factors other than 1. So, we proceed to the next step.

2. Recognize as a Perfect Square Trinomial

Next, we need to recognize that the given quadratic expression is a perfect square trinomial. A perfect square trinomial can be written in the form $(Ay + B)^2 = A^2y^2 + 2ABY + B^2$. Let's verify if our expression fits this form:

• The first term, $25y^2$, is a perfect square: $(5y)^2 = 25y^2$
• The last term, $144$, is also a perfect square: $(12)^2 = 144$
• The middle term, $120y$, can be expressed as $2 \cdot (5y) \cdot (12) = 120y$

Since all three conditions are met, $25y^2 + 120y + 144$ is indeed a perfect square trinomial. This recognition simplifies the factoring process significantly. Perfect square trinomials are always easier to factor because they follow a predictable pattern. Identifying this pattern early on can save time and reduce the chances of making mistakes. Now that we've confirmed it's a perfect square trinomial, we can move on to the next step to express it in its factored form.

3. Express in Factored Form

Since we have identified that $25y^2 + 120y + 144$ is a perfect square trinomial, we can express it in the form $(Ay + B)^2$. In this case, $A = 5$ and $B = 12$. Therefore, the factored form of the quadratic expression is:

$(5y + 12)^2$

This can also be written as:

$(5y + 12)(5y + 12)$

Thus, the completely factored form of the given quadratic expression is $(5y + 12)(5y + 12)$, or $(5y + 12)^2$. This final form represents the simplified and factored version of the original expression. By recognizing the perfect square trinomial pattern, we were able to efficiently factor the expression without needing to use more complex methods. This approach highlights the importance of pattern recognition in factoring quadratic expressions.

Alternative Method: Using the AC Method

If you didn't immediately recognize the perfect square trinomial, you could still factor the expression using the AC method. Let's walk through this alternative approach to reinforce our understanding.

1. Multiply A and C

In the quadratic expression $25y^2 + 120y + 144$, $A = 25$ and $C = 144$. Multiply these two values:

$AC = 25 \cdot 144 = 3600$

2. Find Two Numbers

Now, we need to find two numbers that multiply to $3600$ and add up to $B = 120$. These numbers are $60$ and $60$, since:

$60 \cdot 60 = 3600$

$60 + 60 = 120$

3. Rewrite the Middle Term

Rewrite the middle term, $120y$, using the two numbers we found:

$25y^2 + 60y + 60y + 144$

4. Factor by Grouping

Now, factor by grouping the first two terms and the last two terms:

$5y(5y + 12) + 12(5y + 12)$

Notice that $(5y + 12)$ is a common factor. Factor it out:

$(5y + 12)(5y + 12)$

5. Final Factored Form

The factored form of the quadratic expression is:

$(5y + 12)(5y + 12)$ or $(5y + 12)^2$

As you can see, the AC method leads us to the same result as recognizing the perfect square trinomial. This method is particularly useful when the perfect square trinomial pattern isn't immediately apparent. The AC method provides a systematic way to factor quadratic expressions, ensuring you can handle a wide range of problems. Both methods are valuable tools in your factoring arsenal, and choosing the right one often depends on the specific problem and your personal preference. Now that we've explored both methods, you have a solid understanding of how to factor this type of quadratic expression.

Conclusion

In summary, we successfully factored the quadratic expression $25y^2 + 120y + 144$ completely and simplified it to $(5y + 12)^2$. We achieved this by recognizing the expression as a perfect square trinomial, which allowed for a straightforward factoring process. Additionally, we demonstrated the AC method as an alternative approach, reinforcing the result. Understanding different factoring techniques is crucial for mastering algebra. By practicing these methods, you'll become more confident and efficient in factoring quadratic expressions. Remember, whether you spot the perfect square trinomial or prefer the AC method, the goal is to break down the expression into its simplest factored form. Keep practicing, and you'll find factoring becomes second nature. Factoring is a fundamental skill in mathematics, and mastering it will undoubtedly benefit you in more advanced topics. So, keep honing your skills and enjoy the process of solving these mathematical puzzles!