Which Is The Completely Factored Form Of 4x2 28x 49

Factoring Quadratic Expressions: A Deep Dive into 4x² + 28x + 49

Factoring quadratic expressions is a fundamental skill in algebra. It's the process of rewriting a quadratic expression (an expression of the form ax² + bx + c, where a, b, and c are constants) as a product of two or more simpler expressions. Mastering this skill is crucial for solving quadratic equations, simplifying complex algebraic expressions, and understanding various mathematical concepts. This article will delve deep into factoring the specific quadratic expression 4x² + 28x + 49, exploring various methods and emphasizing the underlying principles.

Understanding Quadratic Expressions

Before we tackle the specific problem, let's review the basics of quadratic expressions. The general form is ax² + bx + c, where:

• a is the coefficient of the x² term (the quadratic term).
• b is the coefficient of the x term (the linear term).
• c is the constant term.

The goal of factoring is to find two binomials (expressions with two terms) whose product is equal to the original quadratic expression. This factorization simplifies the expression and often makes it easier to solve equations or perform other algebraic manipulations.

Methods for Factoring Quadratic Expressions

Several methods exist for factoring quadratic expressions. The most common ones include:

• Greatest Common Factor (GCF) Method: This method involves finding the greatest common factor among all the terms and factoring it out.
• Trial and Error Method: This method involves systematically trying different pairs of binomial factors until you find the correct combination.
• AC Method (also known as the grouping method): This method is a more systematic approach, especially useful when the coefficients are not easily factorable by trial and error.
• Quadratic Formula: While not strictly a factoring method, the quadratic formula can be used to find the roots of the quadratic equation (ax² + bx + c = 0), which can then be used to factor the expression.

Factoring 4x² + 28x + 49: A Step-by-Step Approach

Now, let's focus on factoring the specific expression: 4x² + 28x + 49. We'll explore different methods and highlight why the chosen method is most efficient in this case.

1. Checking for a Greatest Common Factor (GCF)

The first step in factoring any expression is to check for a greatest common factor (GCF) among all the terms. In this case, the coefficients 4, 28, and 49 have a common factor of 1. Therefore, there is no GCF to factor out.

2. Recognizing a Perfect Square Trinomial

Observe the expression: 4x² + 28x + 49. Notice the following:

• 4x² is the square of 2x ( (2x)² = 4x² ).
• 49 is the square of 7 ( 7² = 49 ).
• The middle term, 28x, is twice the product of 2x and 7 ( 2 * (2x) * 7 = 28x ).

This pattern signifies a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form is a² + 2ab + b² = (a + b)².

3. Factoring the Perfect Square Trinomial

Based on the pattern of a perfect square trinomial, we can directly factor 4x² + 28x + 49 as follows:

4x² + 28x + 49 = (2x + 7)²

Therefore, the completely factored form of 4x² + 28x + 49 is (2x + 7)(2x + 7) or (2x + 7)².

Alternative Methods: Illustrating the AC Method

While the perfect square trinomial recognition was the most efficient method in this case, let's illustrate the AC method to demonstrate its applicability to other quadratic expressions, even though it's less efficient for perfect square trinomials.

1. Identify a, b, and c

In the expression 4x² + 28x + 49, we have:

• a = 4
• b = 28
• c = 49

2. Calculate ac

ac = 4 * 49 = 196

3. Find two numbers that add up to b and multiply to ac

We need to find two numbers that add up to 28 and multiply to 196. These numbers are 14 and 14 (14 + 14 = 28 and 14 * 14 = 196).

4. Rewrite the expression using the two numbers

Rewrite the middle term (28x) using the two numbers we found:

4x² + 14x + 14x + 49

5. Factor by grouping

Group the terms in pairs and factor out the GCF from each pair:

(4x² + 14x) + (14x + 49)

2x(2x + 7) + 7(2x + 7)

6. Factor out the common binomial

Notice that both terms now have a common factor of (2x + 7):

(2x + 7)(2x + 7) = (2x + 7)²

This confirms that the completely factored form is indeed (2x + 7)².

Applications of Factoring Quadratic Expressions

Understanding how to factor quadratic expressions is crucial for various mathematical applications, including:

• Solving Quadratic Equations: Factoring allows you to easily solve quadratic equations by setting each factor equal to zero and solving for x. For example, if (2x + 7)² = 0, then 2x + 7 = 0, which gives x = -7/2.

• Simplifying Algebraic Expressions: Factoring can simplify complex algebraic expressions, making them easier to manipulate and understand.

• Graphing Quadratic Functions: The factored form of a quadratic expression reveals the x-intercepts (roots) of the corresponding quadratic function, providing valuable information for graphing the function.

• Calculus: Factoring plays a significant role in calculus, particularly in finding derivatives and integrals.

• Real-world applications: Quadratic equations and their solutions are applied in various real-world scenarios, such as projectile motion, area calculations, and optimization problems.

Conclusion: Mastering the Art of Factoring

Factoring quadratic expressions is a foundational algebraic skill with broad applications. While various methods exist, choosing the most efficient method depends on the characteristics of the specific quadratic expression. Understanding the concept of perfect square trinomials, as demonstrated with 4x² + 28x + 49, is invaluable. Consistent practice and a solid understanding of the underlying principles are key to mastering this crucial skill and unlocking its practical applications in various areas of mathematics and beyond. Remember to always check for a GCF as a first step and carefully examine the coefficients to see if simpler methods, such as recognizing a perfect square trinomial, can be used. If not, then methods like the AC method provide a systematic approach to arrive at the factored form.