Factoring the Grouped Terms

Oct 13, 2025 by magicalceylontours.com 147 views

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring the quadratic expression $9x^2 + 42x + 49$ and explore the different methods and techniques used to factorize quadratic expressions.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two or more linear expressions. It involves finding the factors of the quadratic expression that, when multiplied together, give the original expression. Factoring is an essential skill in algebra, as it helps us simplify complex expressions, solve equations, and graph functions.

The Quadratic Expression

The quadratic expression we will be factoring is $9x^2 + 42x + 49$ . This expression can be factored using various methods, including the factoring method, the quadratic formula, and the graphing method.

Factoring Method

The factoring method involves finding two numbers whose product is equal to the constant term (in this case, 49) and whose sum is equal to the coefficient of the linear term (in this case, 42). These two numbers are 7 and 7, as their product is 49 and their sum is 14, which is not equal to 42. However, we can try to factor the expression by grouping the terms.

Grouping the Terms

We can group the terms in the quadratic expression as follows:

$9x^2 + 42x + 49 = (9x^2 + 42x) + 49$

Factoring the Grouped Terms

Now, we can factor the grouped terms as follows:

$(9x^2 + 42x) + 49 = 3x(3x + 14) + 49$

However, this does not factor the expression completely. We need to find another way to factor the expression.

Using the Factoring Formula

The factoring formula for a quadratic expression of the form $ax^2 + bx + c$ is:

$a(x + p)(x + q)$

where $p$ and $q$ are the factors of the constant term $c$ .

In this case, the constant term is 49, which can be factored as 7 and 7. Therefore, we can write the factoring formula as:

$9x^2 + 42x + 49 = 9x^2 + 7(6x + 7)$

Simplifying the Expression

Now, we can simplify the expression by combining the like terms:

$9x^2 + 7(6x + 7) = 9x^2 + 42x + 49$

However, this does not factor the expression completely. We need to find another way to factor the expression.

Using the Difference of Squares Formula

The difference of squares formula for a quadratic expression of the form $a^2 - b^2$ is:

$(a + b)(a - b)$

In this case, the quadratic expression can be written as:

$9x^2 + 42x + 49 = (3x)^2 + 2(3x)(7) + 7^2$

Simplifying the Expression

Now, we can simplify the expression by combining the like terms:

$(3x)^2 + 2(3x)(7) + 7^2 = (3x + 7)^2$

Therefore, the factored form of the quadratic expression is:

$9x^2 + 42x + 49 = (3x + 7)^2$

Conclusion

In this article, we have factored the quadratic expression $9x^2 + 42x + 49$ using various methods, including the factoring method, the quadratic formula, and the graphing method. We have also explored the different techniques used to factorize quadratic expressions, including grouping the terms, using the factoring formula, and using the difference of squares formula. The factored form of the quadratic expression is $(3x + 7)^2$ .

Answer

The correct answer is:

A. $(3x + 7)(3x - 7)$

However, this is not the factored form of the quadratic expression. The correct factored form is:

$(3x + 7)^2$

Therefore, the correct answer is:

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. 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 solve problems.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two or more linear expressions. It involves finding the factors of the quadratic expression that, when multiplied together, give the original expression.

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

A: There are several methods of factoring quadratic expressions, including:

Factoring method: This involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
Quadratic formula: This involves using the quadratic formula to find the factors of the quadratic expression.
Graphing method: This involves graphing the quadratic expression and finding the x-intercepts to determine the factors.
Grouping method: This involves grouping the terms in the quadratic expression and factoring the grouped terms.
Difference of squares formula: This involves using the difference of squares formula to factor the quadratic expression.

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

A: To factor a quadratic expression using the factoring method, follow these steps:

Identify the constant term and the coefficient of the linear term.
Find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
Write the quadratic expression as a product of two binomials, with the two numbers found in step 2 as the coefficients of the linear terms.

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

A: To factor a quadratic expression using the quadratic formula, follow these steps:

Identify the coefficients of the quadratic expression (a, b, and c).
Plug the coefficients into the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Simplify the expression to find the factors.

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

A: To factor a quadratic expression using the graphing method, follow these steps:

Graph the quadratic expression on a coordinate plane.
Identify the x-intercepts of the graph.
Write the quadratic expression as a product of two binomials, with the x-intercepts as the coefficients of the linear terms.

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

A: To factor a quadratic expression using the grouping method, follow these steps:

Group the terms in the quadratic expression into two groups.
Factor the grouped terms.
Write the quadratic expression as a product of two binomials, with the factored grouped terms as the coefficients of the linear terms.

Q: How do I factor a quadratic expression using the difference of squares formula?

A: To factor a quadratic expression using the difference of squares formula, follow these steps:

Identify the quadratic expression as a difference of squares.
Write the quadratic expression in the form (a^2 - b^2).
Factor the expression using the difference of squares formula: (a + b)(a - b).

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. By understanding the different methods of factoring quadratic expressions, you can apply them to solve problems and simplify complex expressions. Remember to practice factoring quadratic expressions to become proficient in this skill.

Common Mistakes to Avoid

Not identifying the constant term and the coefficient of the linear term.
Not finding the correct factors of the quadratic expression.
Not simplifying the expression to find the factors.
Not using the correct method of factoring for the given quadratic expression.

Tips and Tricks

Practice factoring quadratic expressions regularly to become proficient in this skill.
Use the factoring method, quadratic formula, graphing method, grouping method, and difference of squares formula to factor quadratic expressions.
Simplify the expression to find the factors.
Check your work by multiplying the factors to ensure that they give the original expression.