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 simpler expressions. This skill is crucial for solving quadratic equations, simplifying rational expressions, and understanding many other algebraic concepts. This article will explore the complete factoring of the quadratic expression 4x² + 28x + 49, providing a comprehensive understanding of the process and its underlying principles.

Understanding Quadratic Expressions

Before diving into the factoring process, let's review the basics of quadratic expressions. A quadratic expression is characterized by its highest power of x being 2. The general form is ax² + bx + c, where:

a: The coefficient of the x² term (the quadratic term).
b: The coefficient of the x term (the linear term).
c: The constant term.

In our case, 4x² + 28x + 49, we have a = 4, b = 28, and c = 49.

Methods for Factoring Quadratic Expressions

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

Greatest Common Factor (GCF) Method: This involves finding the largest factor common to all terms and factoring it out.
Factoring by Grouping: This method is useful when the quadratic expression has four or more terms.
Perfect Square Trinomial Method: This method applies when the quadratic expression is a perfect square trinomial—a trinomial that can be factored into the square of a binomial.
AC Method (also known as the "splitting the middle term" method): This is a general method that works for most quadratic expressions.

Factoring 4x² + 28x + 49: Identifying the Perfect Square Trinomial

The expression 4x² + 28x + 49 is a special case of a quadratic expression—it's a perfect square trinomial. A perfect square trinomial is a trinomial that results from squaring a binomial. The general form is:

(a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b²

Let's examine if 4x² + 28x + 49 fits this pattern:

Check the first term: 4x² is the square of 2x ( (2x)² = 4x² ).
Check the last term: 49 is the square of 7 (7² = 49).
Check the middle term: The middle term, 28x, should be twice the product of the terms we found in steps 1 and 2. Indeed, 2 * (2x) * 7 = 28x.

Since all three conditions are met, 4x² + 28x + 49 is a perfect square trinomial.

The Completely Factored Form

Because 4x² + 28x + 49 is a perfect square trinomial, it can be factored using the formula (a + b)² = a² + 2ab + b². In this case, a = 2x and b = 7. Therefore, the completely factored form is:

(2x + 7)²

This means that (2x + 7)(2x + 7) = 4x² + 28x + 49. This is the completely factored form because it cannot be further simplified.

Expanding the Factored Form to Verify the Solution

To verify our solution, we can expand the factored form (2x + 7)² using the FOIL method (First, Outer, Inner, Last):

First: (2x)(2x) = 4x²
Outer: (2x)(7) = 14x
Inner: (7)(2x) = 14x
Last: (7)(7) = 49

Adding these terms together: 4x² + 14x + 14x + 49 = 4x² + 28x + 49. This confirms that our factored form is correct.

Why Understanding Factoring is Crucial

The ability to factor quadratic expressions is vital for numerous algebraic applications:

Solving Quadratic Equations: The factored form of a quadratic equation allows you to easily find its roots (solutions) by setting each factor to zero. For instance, to solve 4x² + 28x + 49 = 0, we would set (2x + 7)² = 0, leading to 2x + 7 = 0, and thus x = -7/2. This is a repeated root because the quadratic is a perfect square.
Simplifying Rational Expressions: Factoring allows you to simplify rational expressions (fractions with polynomials in the numerator and denominator) by canceling common factors.
Graphing Quadratic Functions: The factored form can help determine the x-intercepts (where the graph crosses the x-axis) of a quadratic function.
Solving Real-World Problems: Quadratic equations are used to model many real-world phenomena, including projectile motion, area calculations, and optimization problems. Factoring is essential for solving these types of problems.

Advanced Considerations: Dealing with Different Quadratic Forms

While the perfect square trinomial method worked perfectly for 4x² + 28x + 49, let's explore how other methods could be applied to similar but slightly different quadratic expressions. This will help to broaden your understanding and mastery of quadratic factoring.

1. Using the AC Method: The AC method, or splitting the middle term, is a more general approach. It works for all factorable quadratic expressions, including perfect square trinomials.

Let's use it for the expression 4x² + 28x + 49:

Find AC: In our expression, a = 4, b = 28, and c = 49. Therefore, AC = 4 * 49 = 196.
Find two numbers that add up to b (28) and multiply to AC (196): These numbers are 14 and 14 (14 + 14 = 28 and 14 * 14 = 196).
Rewrite the middle term: We rewrite 28x as 14x + 14x. Our expression now becomes 4x² + 14x + 14x + 49.
Factor by grouping: We group the terms in pairs: (4x² + 14x) + (14x + 49).
Factor out the GCF from each pair: 2x(2x + 7) + 7(2x + 7).
Factor out the common binomial: (2x + 7)(2x + 7) = (2x + 7)².

As you can see, even though we used a different method, we arrived at the same completely factored form.

2. Dealing with Expressions that Aren't Perfect Square Trinomials:

Let's consider a slightly different expression: 4x² + 26x + 49. This is not a perfect square trinomial because the middle term doesn't fit the pattern of 2ab. In such cases, we need to utilize methods like the AC method or trial and error. Using the AC method:

AC = 4 * 49 = 196
We need two numbers that add to 26 and multiply to 196. No such integers exist!

This indicates that 4x² + 26x + 49 is a prime quadratic expression, meaning it cannot be factored using integer coefficients. In these situations, the expression is considered to be in its simplest factored form. It's important to recognize when an expression is prime, as trying to force a factorization where one doesn't exist can lead to errors.

Conclusion: Mastering Quadratic Factoring

Factoring quadratic expressions is a cornerstone of algebra. Understanding the various methods and recognizing patterns like perfect square trinomials is crucial for success in algebra and related fields. While the perfect square trinomial method provided a straightforward solution for 4x² + 28x + 49, learning and applying other methods, such as the AC method, enhances your problem-solving skills and expands your ability to tackle a broader range of quadratic expressions. Remember, practice is key to mastering this important skill. The more you practice, the more efficiently you will be able to identify the appropriate method and factor various quadratic expressions.