Factor Completely:$\[ 5x^2 - 19x - 4 \\]Answer Attempt 1 Out Of 2:\[$\square\$\]

Feb 26, 2025 by ADMIN 81 views

Factoring a Quadratic Expression: A Step-by-Step Guide

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Introduction

Factoring a quadratic expression is a fundamental concept in algebra that involves expressing the given expression as a product of two or more polynomials. In this article, we will focus on factoring the quadratic expression $5x^2 - 19x - 4$ . Factoring quadratic expressions is an essential skill that is used in various mathematical applications, including solving equations, graphing functions, and analyzing data.

Understanding the Basics of Factoring

Before we dive into factoring the given quadratic expression, it's essential to understand the basics of factoring. Factoring involves expressing an expression as a product of two or more polynomials. The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. To factor a quadratic expression, we need to find two binomials whose product is equal to the given expression.

The Factoring Process

The factoring process involves several steps:

Identify the greatest common factor (GCF): The first step in factoring a quadratic expression is to identify the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides all the terms evenly.
Write the GCF as a factor: Once we have identified the GCF, we can write it as a factor of the expression.
Distribute the GCF: After writing the GCF as a factor, we need to distribute it to each term in the expression.
Factor the remaining expression: After distributing the GCF, we are left with a quadratic expression that can be factored further.

Factoring the Given Quadratic Expression

Now that we have understood the basics of factoring and the factoring process, let's apply these concepts to factor the given quadratic expression $5x^2 - 19x - 4$ .

Step 1: Identify the GCF

The first step in factoring the given quadratic expression is to identify the greatest common factor (GCF) of the terms. In this case, the GCF is 1, since there is no common factor that divides all the terms evenly.

Step 2: Write the GCF as a factor

Since the GCF is 1, we can write it as a factor of the expression. The expression can be written as $1(5x^2 - 19x - 4)$ .

Step 3: Distribute the GCF

After writing the GCF as a factor, we need to distribute it to each term in the expression. However, since the GCF is 1, we don't need to distribute it.

Step 4: Factor the remaining expression

After distributing the GCF, we are left with the quadratic expression $5x^2 - 19x - 4$ . To factor this expression, we need to find two binomials whose product is equal to the given expression.

Factoring by Grouping

One method of factoring a quadratic expression is by grouping. This involves grouping the terms in pairs and factoring out the GCF from each pair.

Step 2: Group the terms

To factor the quadratic expression $5x^2 - 19x - 4$ by grouping, we need to group the terms in pairs. We can group the terms as follows:

$(5x^2 - 4x) - (15x - 4)$

Step 3: Factor out the GCF from each pair

After grouping the terms, we need to factor out the GCF from each pair. The GCF of the first pair is $x$ , and the GCF of the second pair is $-1$ .

$x(5x - 4) - 1(15x - 4)$

Step 4: Factor the remaining expression

After factoring out the GCF from each pair, we are left with the quadratic expression $x(5x - 4) - 1(15x - 4)$ . To factor this expression, we need to find two binomials whose product is equal to the given expression.

Factoring by Using the AC Method

Another method of factoring a quadratic expression is by using the AC method. This involves factoring the expression as $(ax + b)(cx + d)$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Step 2: Identify the values of a, b, c, and d

To factor the quadratic expression $5x^2 - 19x - 4$ using the AC method, we need to identify the values of $a$ , $b$ , $c$ , and $d$ . We can choose $a = 5$ , $b = -4$ , $c = -1$ , and $d = 1$ .

Step 3: Write the expression as a product of two binomials

After identifying the values of $a$ , $b$ , $c$ , and $d$ , we can write the expression as a product of two binomials:

$(5x - 4)(-x + 1)$

Step 4: Simplify the expression

After writing the expression as a product of two binomials, we can simplify it by multiplying the two binomials:

$-5x^2 + 5x + 4x - 4$

$-5x^2 + 9x - 4$

Conclusion

In this article, we have learned how to factor a quadratic expression using various methods, including factoring by grouping and using the AC method. We have also applied these concepts to factor the given quadratic expression $5x^2 - 19x - 4$ . Factoring quadratic expressions is an essential skill that is used in various mathematical applications, including solving equations, graphing functions, and analyzing data.

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing the given expression as a product of two or more polynomials. In this article, we will provide a Q&A guide to help you understand the basics of factoring quadratic expressions and how to apply these concepts to solve problems.

Q&A: Factoring Quadratic Expressions

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves expressing the given expression as a product of two or more polynomials.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including factoring by grouping, using the AC method, and factoring by using the greatest common factor (GCF).

Q: How do I factor a quadratic expression using the GCF method?

A: To factor a quadratic expression using the GCF method, you need to identify the greatest common factor (GCF) of the terms and write it as a factor of the expression.

Q: How do I factor a quadratic expression using the AC method?

A: To factor a quadratic expression using the AC method, you need to identify the values of a, b, c, and d and write the expression as a product of two binomials.

Q: What is the difference between factoring by grouping and using the AC method?

A: Factoring by grouping involves grouping the terms in pairs and factoring out the GCF from each pair, while using the AC method involves factoring the expression as (ax + b)(cx + d).

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to check if the expression can be written as a product of two or more polynomials.

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

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

Not identifying the GCF correctly
Not distributing the GCF correctly
Not factoring the expression correctly
Not checking if the expression can be factored

Q: How do I check if a factored expression is correct?

A: To check if a factored expression is correct, you need to multiply the two binomials and simplify the expression to see if it matches the original expression.

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

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

Solving equations
Graphing functions
Analyzing data
Modeling real-world problems

Conclusion

In this article, we have provided a Q&A guide to help you understand the basics of factoring quadratic expressions and how to apply these concepts to solve problems. Factoring quadratic expressions is an essential skill that is used in various mathematical applications, including solving equations, graphing functions, and analyzing data.

Additional Resources

Final Thoughts

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing the given expression as a product of two or more polynomials. By understanding the basics of factoring quadratic expressions and how to apply these concepts to solve problems, you can develop a strong foundation in algebra and apply these skills to real-world problems.