Select The Correct Answer.Which Expression Shows The Factors Of $5x^2 + 17x + 6$?A. $(5x + 1)(x + 6)$ B. $ ( 5 X + 2 ) ( X + 3 ) (5x + 2)(x + 3) ( 5 X + 2 ) ( X + 3 ) [/tex] C. $(5x + 3)(x + 2)$ D. $(5x + 6)(x + 1)$

Mar 13, 2025 by ADMIN 227 views

**Factoring Quadratic Expressions: A Step-by-Step Guide**

===========================================================

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we will explore the process of factoring quadratic expressions and apply it to a specific problem to determine the correct answer.

What are Quadratic Expressions?

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Quadratic expressions can be factored into the product of two binomials, which is a fundamental concept in algebra.

How to Factor Quadratic Expressions

To factor a quadratic expression, we need to find two numbers whose product is equal to the constant term ( $c$ ) and whose sum is equal to the coefficient of the linear term ( $b$ ). These two numbers are called the factors of the quadratic expression. Once we have found the factors, we can write the quadratic expression as the product of two binomials.

The Problem

The problem asks us to select the correct answer, which shows the factors of the quadratic expression $5x^2 + 17x + 6$ . We have four options to choose from:

A. $(5x + 1)(x + 6)$ B. $(5x + 2)(x + 3)$ C. $(5x + 3)(x + 2)$ D. $(5x + 6)(x + 1)$

Step 1: Find the Factors

To find the factors of the quadratic expression, we need to find two numbers whose product is equal to the constant term ( $6$ ) and whose sum is equal to the coefficient of the linear term ( $17$ ). Let's call these two numbers $m$ and $n$ . We can write the following equations:

$m \times n = 6$ $m + n = 17$

Step 2: Solve the Equations

We can solve the equations by trial and error or by using algebraic methods. Let's try to find the factors by trial and error. We can start by listing the factors of $6$ and checking if their sum is equal to $17$ .

The factors of $6$ are: $1, 2, 3, 6$

We can check if the sum of each pair of factors is equal to $17$ :

$1 + 6 = 7$ (not equal to $17$ )
$2 + 3 = 5$ (not equal to $17$ )
$1 + 2 = 3$ (not equal to $17$ )
$2 + 6 = 8$ (not equal to $17$ )
$3 + 6 = 9$ (not equal to $17$ )

However, we can see that $5x + 2$ and $x + 3$ are factors of the quadratic expression, since their product is equal to $5x^2 + 17x + 6$ .

Conclusion

Based on the analysis above, we can conclude that the correct answer is:

B. $(5x + 2)(x + 3)$

This is because the product of the two binomials is equal to the quadratic expression, and the sum of the coefficients of the linear terms is equal to the coefficient of the linear term in the quadratic expression.

Final Answer

The final answer is:

B. $(5x + 2)(x + 3)$

=====================================================

Introduction

In our previous article, we explored the process of factoring quadratic expressions and applied it to a specific problem to determine the correct answer. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to different problems.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is equal to the constant term ( $c$ ) and whose sum is equal to the coefficient of the linear term ( $b$ ). These two numbers are called the factors of the quadratic expression. Once you have found the factors, you can write the quadratic expression as the product of two binomials.

Q: What are the steps to factor a quadratic expression?

A: The steps to factor a quadratic expression are:

Find the factors of the constant term ( $c$ ).
Check if the sum of each pair of factors is equal to the coefficient of the linear term ( $b$ ).
Write the quadratic expression as the product of two binomials using the factors you found.

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

A: A quadratic expression can be factored if it can be written as the product of two binomials. To check if a quadratic expression can be factored, you can try to find the factors of the constant term ( $c$ ) and check if the sum of each pair of factors is equal to the coefficient of the linear term ( $b$ ).

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

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

Not checking if the sum of each pair of factors is equal to the coefficient of the linear term ( $b$ ).
Not using the correct factors of the constant term ( $c$ ).
Not writing the quadratic expression as the product of two binomials.

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

A: To check if a quadratic expression is factored correctly, you can multiply the two binomials and check if the result is equal to the original quadratic expression.

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 problems using quadratic equations

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. By understanding the concept of factoring quadratic expressions and how to apply it to different problems, you can become a more confident and proficient math student.

Final Tips

Practice factoring quadratic expressions regularly to become more comfortable with the concept.
Use online resources or math textbooks to help you understand the concept of factoring quadratic expressions.
Apply factoring quadratic expressions to real-world problems to see the practical applications of the concept.

Common Quadratic Expressions

Here are some common quadratic expressions that can be factored:

$x^2 + 5x + 6$
$x^2 - 7x + 12$
$x^2 + 2x - 15$
$x^2 - 9x + 20$

Factoring Quadratic Expressions Worksheet

Here is a worksheet with some quadratic expressions that can be factored:

Factor the quadratic expression: $x^2 + 5x + 6$
Factor the quadratic expression: $x^2 - 7x + 12$
Factor the quadratic expression: $x^2 + 2x - 15$
Factor the quadratic expression: $x^2 - 9x + 20$

Answer Key

Here is the answer key for the worksheet:

$(x + 3)(x + 2)$
$(x - 3)(x - 4)$
$(x + 5)(x - 3)$
$(x - 5)(x - 4)$