Factor The Expression: ${ 25w^2 - 60w + 36 }$

Introduction

In mathematics, factoring an expression is a fundamental concept that involves breaking down a given expression into a product of simpler expressions. This process is essential in solving equations, graphing functions, and simplifying complex expressions. In this article, we will focus on factoring the expression $25w^2 - 60w + 36$. We will explore various techniques and methods to factor this expression, and provide a step-by-step guide on how to do it.

Understanding the Expression

Before we dive into factoring the expression, let's take a closer look at it. The given expression is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 25$, $b = -60$, and $c = 36$. The expression can be written as:

$$25w^2 - 60w + 36$$

Factoring Techniques

There are several techniques used to factor quadratic expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the expression in pairs and factoring out the common factors.
• Factoring by Difference of Squares: This method involves factoring the expression as the difference of two squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the expression as a perfect square trinomial.

Factoring by Grouping

Let's start by factoring the expression using the grouping method. We can group the terms of the expression as follows:

$$25w^2 - 60w + 36 = (25w^2 - 36) - (60w - 36)$$

Now, we can factor out the common factors from each group:

$$(25w^2 - 36) - (60w - 36) = 25w^2 - 36 - 60w + 36$$

We can simplify the expression by combining like terms:

$$25w^2 - 36 - 60w + 36 = 25w^2 - 60w$$

However, we still have a common factor of $25w^2 - 60w$. We can factor out the common factor as follows:

$$25w^2 - 60w = 25w(w - \frac{12}{5})$$

Factoring by Difference of Squares

Another technique used to factor quadratic expressions is the difference of squares method. This method involves factoring the expression as the difference of two squares.

However, the given expression $25w^2 - 60w + 36$ does not fit the form of a difference of squares. Therefore, we cannot use this method to factor the expression.

Factoring by Perfect Square Trinomials

The given expression $25w^2 - 60w + 36$ can be written as a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored as the square of a binomial.

We can rewrite the expression as follows:

$$25w^2 - 60w + 36 = (5w)^2 - 2(5w)(6) + (6)^2$$

Now, we can factor the expression as the square of a binomial:

$$(5w)^2 - 2(5w)(6) + (6)^2 = (5w - 6)^2$$

Conclusion

In this article, we have explored various techniques and methods to factor the expression $25w^2 - 60w + 36$. We have used the grouping method, difference of squares method, and perfect square trinomials method to factor the expression. We have found that the expression can be factored as $(5w - 6)^2$.

Final Answer

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Q&A: Frequently Asked Questions

Q: What is factoring an expression?

A: Factoring an expression is a process of breaking down a given expression into a product of simpler expressions. This process is essential in solving equations, graphing functions, and simplifying complex expressions.

Q: Why is factoring an expression important?

A: Factoring an expression is important because it allows us to simplify complex expressions, solve equations, and graph functions. It is a fundamental concept in mathematics and is used in various fields such as physics, engineering, and economics.

Q: What are the different techniques used to factor expressions?

A: There are several techniques used to factor expressions, including:

• Factoring by Grouping: This method involves grouping the terms of the expression in pairs and factoring out the common factors.
• Factoring by Difference of Squares: This method involves factoring the expression as the difference of two squares.
• Factoring by Perfect Square Trinomials: This method involves factoring the expression as a perfect square trinomial.

Q: How do I factor an expression using the grouping method?

A: To factor an expression using the grouping method, you need to group the terms of the expression in pairs and factor out the common factors. For example, if you have the expression $25w^2 - 60w + 36$, you can group the terms as follows:

$$(25w^2 - 36) - (60w - 36)$$

Now, you can factor out the common factors from each group:

$$(25w^2 - 36) - (60w - 36) = 25w^2 - 36 - 60w + 36$$

You can simplify the expression by combining like terms:

$$25w^2 - 36 - 60w + 36 = 25w^2 - 60w$$

However, you still have a common factor of $25w^2 - 60w$. You can factor out the common factor as follows:

$$25w^2 - 60w = 25w(w - \frac{12}{5})$$

Q: How do I factor an expression using the difference of squares method?

A: To factor an expression using the difference of squares method, you need to factor the expression as the difference of two squares. For example, if you have the expression $a^2 - b^2$, you can factor it as follows:

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

However, the given expression $25w^2 - 60w + 36$ does not fit the form of a difference of squares. Therefore, you cannot use this method to factor the expression.

Q: How do I factor an expression using the perfect square trinomials method?

A: To factor an expression using the perfect square trinomials method, you need to factor the expression as a perfect square trinomial. For example, if you have the expression $a^2 + 2ab + b^2$, you can factor it as follows:

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

You can rewrite the expression $25w^2 - 60w + 36$ as a perfect square trinomial as follows:

$$(5w)^2 - 2(5w)(6) + (6)^2$$

Now, you can factor the expression as the square of a binomial:

$$(5w)^2 - 2(5w)(6) + (6)^2 = (5w - 6)^2$$

Q: What is the final answer to the expression $25w^2 - 60w + 36$?

A: The final answer to the expression $25w^2 - 60w + 36$ is $\boxed{(5w - 6)^2}$.

Conclusion

In this article, we have explored various techniques and methods to factor the expression $25w^2 - 60w + 36$. We have used the grouping method, difference of squares method, and perfect square trinomials method to factor the expression. We have found that the expression can be factored as $(5w - 6)^2$.