Factoring Quadratics Unveiling The Completely Factored Form Of 4x² + 28x + 49

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In the realm of algebra, factoring quadratic expressions is a fundamental skill. It involves breaking down a quadratic expression into its constituent linear factors, revealing the roots or solutions of the equation. One particular type of quadratic expression, the perfect square trinomial, exhibits a unique pattern that simplifies the factoring process. In this article, we will embark on a journey to unravel the completely factored form of the quadratic expression 4x² + 28x + 49, exploring the concepts of factoring, perfect square trinomials, and their applications.

Understanding Factoring and Quadratic Expressions

Before we delve into the specific problem at hand, let's lay a foundation by understanding the core concepts of factoring and quadratic expressions. Factoring, in essence, is the reverse process of expanding algebraic expressions. When we expand, we multiply expressions together to obtain a larger expression. Factoring, on the other hand, involves breaking down a larger expression into its constituent factors, which are expressions that, when multiplied together, yield the original expression.

Quadratic expressions are algebraic expressions of the form ax² + bx + c, where a, b, and c are constants, and x is a variable. The term 'quadratic' stems from the fact that the highest power of the variable x is 2. Quadratic expressions are ubiquitous in mathematics and physics, appearing in various contexts such as projectile motion, optimization problems, and curve fitting.

Factoring quadratic expressions is a crucial skill in algebra, as it allows us to solve quadratic equations, simplify expressions, and analyze the behavior of quadratic functions. Several techniques exist for factoring quadratic expressions, including trial and error, grouping, and the use of special patterns such as the difference of squares and perfect square trinomials.

Recognizing Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that arises from squaring a binomial. A binomial is an algebraic expression consisting of two terms, such as (x + y) or (2x - 3). When we square a binomial, we obtain a trinomial, which is an expression with three terms. A perfect square trinomial exhibits a distinct pattern that makes it easily recognizable and factorable.

The general form of a perfect square trinomial is a² + 2ab + b² or a² - 2ab + b², where a and b are constants or expressions. These expressions correspond to the squares of the binomials (a + b) and (a - b), respectively. Let's examine the pattern more closely:

  • (a + b)² = (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b²
  • (a - b)² = (a - b)(a - b) = a² - ab - ba + b² = a² - 2ab + b²

From these expansions, we can observe the key characteristics of a perfect square trinomial:

  1. The first and last terms are perfect squares: a² and b² are the squares of a and b, respectively.
  2. The middle term is twice the product of the square roots of the first and last terms: 2ab is twice the product of a and b.

Recognizing these patterns allows us to quickly identify perfect square trinomials and factor them efficiently.

Factoring 4x² + 28x + 49: A Perfect Square Trinomial

Now that we have a solid understanding of factoring and perfect square trinomials, let's turn our attention to the specific expression at hand: 4x² + 28x + 49. Our goal is to determine the completely factored form of this expression.

To begin, we must first ascertain whether the given expression is indeed a perfect square trinomial. Let's examine the terms:

  • The first term, 4x², is a perfect square, as it is the square of 2x: (2x)² = 4x².
  • The last term, 49, is also a perfect square, as it is the square of 7: 7² = 49.
  • The middle term, 28x, appears to fit the pattern of twice the product of the square roots of the first and last terms. Let's verify: 2 * (2x) * 7 = 28x.

Since all three conditions are met, we can confidently conclude that 4x² + 28x + 49 is a perfect square trinomial. This means it can be factored into the square of a binomial.

To determine the binomial, we take the square roots of the first and last terms and connect them with the sign of the middle term. In this case:

  • The square root of 4x² is 2x.
  • The square root of 49 is 7.
  • The middle term is positive, so we use the addition sign.

Therefore, the binomial is (2x + 7). Since the expression is a perfect square trinomial, we square this binomial to obtain the factored form:

4x² + 28x + 49 = (2x + 7)² = (2x + 7)(2x + 7)

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

Analyzing the Answer Choices

Now, let's compare our factored form with the given answer choices:

A. (x + 7)(4x + 7) B. 4(x + 7)(x + 7) C. (2x + 7)(2x + 7) D. 2(x + 7)(x + 7)

Clearly, answer choice C. (2x + 7)(2x + 7) matches our derived factored form. Therefore, option C is the correct answer.

The other answer choices are incorrect because they do not represent the correct factorization of the given quadratic expression. Option A is a product of two different binomials, while options B and D include incorrect constant factors.

Applications of Factoring Perfect Square Trinomials

Factoring perfect square trinomials is not merely an academic exercise; it has practical applications in various mathematical and scientific contexts. Here are a few examples:

  1. Solving Quadratic Equations: Factoring is a powerful technique for solving quadratic equations, which are equations of the form ax² + bx + c = 0. If the quadratic expression on the left-hand side is a perfect square trinomial, we can factor it and set the binomial equal to zero to find the solutions. For instance, the equation 4x² + 28x + 49 = 0 can be solved by factoring it as (2x + 7)² = 0, which gives us the solution x = -7/2.
  2. Simplifying Algebraic Expressions: Factoring can be used to simplify complex algebraic expressions by canceling common factors. If a perfect square trinomial appears in the numerator or denominator of a fraction, factoring it can help reduce the fraction to its simplest form.
  3. Graphing Quadratic Functions: The factored form of a quadratic expression provides valuable information about the graph of the corresponding quadratic function. The roots of the equation, which are the values of x that make the expression equal to zero, correspond to the x-intercepts of the graph. If the quadratic expression is a perfect square trinomial, the graph will touch the x-axis at only one point, indicating a double root.
  4. Completing the Square: The technique of completing the square, which is used to rewrite a quadratic expression in vertex form, relies on the concept of perfect square trinomials. By adding a constant term to a quadratic expression, we can transform it into a perfect square trinomial, making it easier to analyze and graph.

Conclusion: Mastering Factoring and Perfect Square Trinomials

In this comprehensive exploration, we have delved into the world of factoring quadratic expressions, with a particular focus on perfect square trinomials. We have learned that factoring is the process of breaking down an expression into its constituent factors, and that quadratic expressions are algebraic expressions of the form ax² + bx + c. Perfect square trinomials, a special type of quadratic expression, exhibit a distinct pattern that simplifies the factoring process.

By recognizing the characteristics of a perfect square trinomial – the first and last terms being perfect squares, and the middle term being twice the product of their square roots – we can efficiently factor them into the square of a binomial. In the case of 4x² + 28x + 49, we successfully identified it as a perfect square trinomial and factored it into (2x + 7)(2x + 7).

Factoring perfect square trinomials is not just a mathematical exercise; it has practical applications in solving quadratic equations, simplifying algebraic expressions, graphing quadratic functions, and completing the square. Mastering these techniques empowers us to tackle a wide range of mathematical problems and gain a deeper understanding of the behavior of quadratic expressions.

As you continue your mathematical journey, remember that practice is key to mastering factoring and other algebraic skills. By working through various examples and problems, you will develop your intuition and become more adept at recognizing patterns and applying the appropriate techniques. With dedication and perseverance, you will unlock the power of factoring and its myriad applications in mathematics and beyond.