Factor The Following Expression Completely:$\[ 5x^2 + 39x + 28 = \\]$\[ \square \\]

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex equations and solve for unknown variables. In this article, we will focus on factoring the given expression $5x^2 + 39x + 28$ completely. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding the Expression

Before we begin factoring, let's take a closer look at the given expression:

$5x^2 + 39x + 28$

This is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 5$, $b = 39$, and $c = 28$. Our goal is to factor this expression completely, which means expressing it as a product of two binomials.

Step 1: Find the Greatest Common Factor (GCF)

The first step in factoring a quadratic expression is to find the greatest common factor (GCF) of the three terms. In this case, the GCF of $5x^2$, $39x$, and $28$ is $1$, since there is no common factor that divides all three terms.

Step 2: Look for Two Numbers Whose Product is $ac$ and Whose Sum is $b$

Now that we have found the GCF, we need to look for two numbers whose product is $ac$ and whose sum is $b$. In this case, $ac = 5 \times 28 = 140$, and $b = 39$. We need to find two numbers whose product is $140$ and whose sum is $39$.

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression is the process of expressing it as a product of two binomials. This involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: How do I start factoring a quadratic expression?

A: To start factoring a quadratic expression, you need to find the greatest common factor (GCF) of the three terms. If there is no common factor, you can proceed to the next step.

Q: What is the next step in factoring a quadratic expression?

A: The next step is to look for two numbers whose product is the product of the coefficient of the quadratic term and the constant term, and whose sum is the coefficient of the linear term.

Q: How do I find the two numbers?

A: To find the two numbers, you can use trial and error or use a formula. One way to find the two numbers is to use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic expression.

Q: What if I don't know how to use the quadratic formula?

A: Don't worry! You can also use trial and error to find the two numbers. Simply try different combinations of numbers until you find two numbers whose product is the product of the coefficient of the quadratic term and the constant term, and whose sum is the coefficient of the linear term.

Q: What if I get stuck?

A: If you get stuck, try breaking down the problem into smaller steps. For example, you can try factoring the quadratic expression by grouping the terms.

Q: What is factoring by grouping?

A: Factoring by grouping is a technique used to factor quadratic expressions. It involves grouping the terms in pairs and then factoring out the greatest common factor of each pair.

Q: How do I factor by grouping?

A: To factor by grouping, follow these steps:

1. Group the terms in pairs.
2. Factor out the greatest common factor of each pair.
3. Combine the two pairs to form a single expression.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

A: Some common mistakes to avoid when factoring quadratic expressions include:

• Not finding the greatest common factor (GCF) of the three terms.
• Not looking for two numbers whose product is the product of the coefficient of the quadratic term and the constant term, and whose sum is the coefficient of the linear term.
• Not using the correct formula or technique to factor the quadratic expression.

Q: How can I practice factoring quadratic expressions?

A: You can practice factoring quadratic expressions by working through examples and exercises. You can also use online resources or worksheets to help you practice.

Q: What are some real-world applications of factoring quadratic expressions?

A: Factoring quadratic expressions has many real-world applications, including:

• Solving systems of equations
• Finding the maximum or minimum value of a function
• Modeling real-world situations, such as the motion of an object under the influence of gravity.

Conclusion

Factoring quadratic expressions is an important concept in algebra that has many real-world applications. By following the steps outlined in this article, you can learn how to factor quadratic expressions and apply this knowledge to solve problems in mathematics and other fields.