Factor The Expression: $4x^2 + 12x - 72$ Using The Form $4(x - ?)(x + 6)$. What Is The Missing Value?

Jun 21, 2025 by ADMIN 102 views

Factoring Quadratic Expressions: A Step-by-Step Guide to Factoring $4x^2 + 12x - 72$

Factoring quadratic expressions is a fundamental skill in algebra, and it's essential for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we will walk through the process of factoring the quadratic expression $4x^2 + 12x - 72$ step by step. This article is written to be SEO-friendly and provide readers with a clear, concise, and valuable resource for mastering factoring techniques.

Understanding the Basics of Factoring

Before we dive into the specifics of factoring $4x^2 + 12x - 72$ , let's review the basic principles of factoring. Factoring is the process of breaking down an expression into its constituent factors, which are terms that, when multiplied together, produce the original expression. In the case of quadratic expressions, we aim to express them as the product of two binomials.

The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. Factoring involves finding two binomials, $(px + q)$ and $(rx + s)$ , such that:

$(px + q)(rx + s) = ax^2 + bx + c$

Our goal is to reverse this process, starting with the quadratic expression and finding the binomial factors. When factoring, several techniques can be employed, such as factoring out the greatest common factor (GCF), using the quadratic formula, or employing the method of grouping. In this article, we will focus on these methods, especially how to identify and factor out the GCF, which is the first crucial step in simplifying any quadratic expression.

The Importance of the Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all terms in an expression. Identifying and factoring out the GCF simplifies the factoring process and makes subsequent steps easier. In the expression $4x^2 + 12x - 72$ , the GCF is a crucial starting point. By recognizing the GCF, we reduce the complexity of the quadratic expression and make it more manageable for further factorization. Factoring out the GCF allows us to work with smaller coefficients, simplifying the factoring process significantly. This initial step not only reduces the size of the numbers involved but also reveals the underlying structure of the quadratic expression, making it easier to apply other factoring techniques. For instance, after factoring out the GCF, the remaining quadratic expression might be a simpler trinomial that is easier to factor using methods such as the quadratic formula or completing the square. Thus, identifying and extracting the GCF is a cornerstone of efficient and accurate factoring, making it an indispensable skill in algebra.

Step 1: Identify the Greatest Common Factor (GCF)

Our first step in factoring the expression $4x^2 + 12x - 72$ is to identify the greatest common factor (GCF) of the coefficients. The coefficients are 4, 12, and -72. To find the GCF, we look for the largest number that divides evenly into all three coefficients.

The factors of 4 are: 1, 2, and 4.
The factors of 12 are: 1, 2, 3, 4, 6, and 12.
The factors of -72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72 (and their negative counterparts).

The largest number that appears in all three lists is 4. Therefore, the GCF of $4x^2$ , $12x$ , and $-72$ is 4.

Factoring out the GCF: A Detailed Explanation

Once we've identified the GCF, which in this case is 4, we factor it out from each term in the expression. This involves dividing each term by the GCF and writing the expression as a product of the GCF and the resulting terms. The process of factoring out the GCF involves dividing each term of the original expression by the GCF and writing the result in parentheses. For the expression $4x^2 + 12x - 72$ , we divide each term by 4:

$4x^2$ divided by 4 equals $x^2$
$12x$ divided by 4 equals $3x$
$-72$ divided by 4 equals $-18$

By factoring out the GCF, we effectively simplify the expression, making it easier to handle in subsequent factoring steps. This process reduces the coefficients, which in turn, simplifies the task of finding the binomial factors. Factoring out the GCF is akin to unwrapping the expression, revealing its underlying structure and making it more amenable to further manipulation. In essence, this step is a critical prerequisite for applying other factoring techniques, as it often transforms a complex expression into a more manageable form. This not only simplifies the calculations but also minimizes the chances of error, ensuring a more accurate factorization process. By meticulously factoring out the GCF, we lay a solid foundation for successfully factoring the quadratic expression.

Step 2: Factor out the GCF

Now that we've identified the GCF as 4, we factor it out of the expression:

$4x^2 + 12x - 72 = 4(x^2 + 3x - 18)$

This simplifies our expression, and we now need to factor the quadratic expression inside the parentheses: $x^2 + 3x - 18$ .

Understanding the Simplified Quadratic Expression

After factoring out the GCF, we are left with a simplified quadratic expression inside the parentheses, which in this case is $x^2 + 3x - 18$ . This simplified expression is significantly easier to factor than the original, as the coefficients are smaller and the overall complexity is reduced. Understanding the structure of this simplified quadratic expression is crucial for the next steps in the factoring process. The general form of a quadratic expression is $ax^2 + bx + c$ , and in our case, $a = 1$ , $b = 3$ , and $c = -18$ . The goal is to find two numbers that multiply to the constant term ( $c$ ) and add up to the coefficient of the linear term ( $b$ ). By focusing on this simplified form, we can systematically determine the factors that will lead to the correct binomial factorization. This step highlights the importance of the GCF method in simplifying complex expressions, making them more accessible for factorization techniques. The reduced complexity not only aids in the factoring process but also enhances understanding, allowing for a more intuitive approach to solving quadratic equations.

Step 3: Factor the Quadratic Expression Inside the Parentheses

We need to factor the quadratic expression $x^2 + 3x - 18$ . To do this, we look for two numbers that multiply to -18 (the constant term) and add to 3 (the coefficient of the $x$ term).

The pairs of factors of -18 are:

1 and -18
-1 and 18
2 and -9
-2 and 9
3 and -6
-3 and 6

The pair that adds up to 3 is -3 and 6. Therefore, we can rewrite the quadratic expression as:

$x^2 + 3x - 18 = (x - 3)(x + 6)$

Deeper Dive into Finding the Right Factors

To successfully factor a quadratic expression of the form $x^2 + bx + c$ , the key is to identify two numbers that satisfy specific conditions related to the coefficients $b$ and $c$ . These two numbers, let’s call them $m$ and $n$ , must multiply to give the value of $c$ (the constant term) and add up to the value of $b$ (the coefficient of the linear term). In the case of our expression $x^2 + 3x - 18$ , we need to find two numbers that multiply to -18 and add up to 3. This systematic approach involves listing the factor pairs of -18 and checking which pair sums to 3. By methodically examining these pairs, we can quickly identify the correct numbers, which are -3 and 6. This process ensures accuracy and efficiency in factoring quadratic expressions. The method is particularly effective because it breaks down the problem into smaller, more manageable steps. Once the correct numbers are identified, they directly translate into the factored form of the quadratic expression, making the process straightforward and less prone to errors. The structured approach not only simplifies the factoring process but also enhances understanding, allowing individuals to apply the method confidently to a variety of quadratic expressions.

Step 4: Combine the Factors

Now we combine the GCF we factored out in Step 2 with the factors we found in Step 3:

$4(x^2 + 3x - 18) = 4(x - 3)(x + 6)$

Therefore, the factored form of $4x^2 + 12x - 72$ is $4(x - 3)(x + 6)$ .

Completing the Factorization: Ensuring Accuracy and Final Form

After factoring the quadratic expression inside the parentheses, the final step is to combine the GCF that was factored out earlier with the binomial factors obtained. This ensures that the factored expression is equivalent to the original quadratic expression. In our case, we factored out a GCF of 4 and obtained the binomial factors $(x - 3)$ and $(x + 6)$ . Combining these gives us the final factored form: $4(x - 3)(x + 6)$ . This final form represents the complete factorization of the original quadratic expression $4x^2 + 12x - 72$ . To ensure accuracy, it is always a good practice to check the factored form by expanding it back to the original expression. Expanding $4(x - 3)(x + 6)$ should yield $4x^2 + 12x - 72$ , confirming that the factorization is correct. The process of combining the factors not only completes the factorization but also provides an opportunity to verify the result, enhancing confidence in the solution. The complete factored form is crucial for various algebraic manipulations, such as solving quadratic equations, simplifying rational expressions, and analyzing graphs of quadratic functions. Therefore, mastering this final step is essential for proficiency in algebra.

Final Answer

The factored form of the expression $4x^2 + 12x - 72$ is $4(x - 3)(x + 6)$ . This matches the requested form $4(x - ext{_})(x + 6)$ , with the missing value being 3.

Summary of Factoring Techniques Used

In this guide, we've demonstrated a step-by-step approach to factoring the quadratic expression $4x^2 + 12x - 72$ . The key techniques we employed include:

Identifying the Greatest Common Factor (GCF): Recognizing and factoring out the GCF simplifies the expression and makes it easier to factor further.
Factoring Quadratic Expressions: We used the method of finding two numbers that multiply to the constant term and add to the coefficient of the linear term.
Combining Factors: We combined the GCF with the binomial factors to obtain the final factored form.

The Importance of Practice and Mastery in Factoring

Mastering factoring techniques is essential for success in algebra and beyond. Factoring is a fundamental skill that underpins many algebraic concepts and is crucial for solving equations, simplifying expressions, and understanding mathematical relationships. The process of factoring quadratic expressions involves breaking down complex polynomials into simpler factors, which is a critical step in solving quadratic equations and analyzing their properties. Moreover, factoring skills are directly applicable in various fields, including engineering, physics, and computer science, where algebraic manipulation is a common task. To achieve proficiency in factoring, consistent practice is key. This involves working through a variety of problems, starting with simpler expressions and gradually progressing to more complex ones. By practicing regularly, individuals can develop an intuitive understanding of the relationships between coefficients and factors, making the factoring process more efficient and accurate. Mastery of factoring not only enhances problem-solving abilities but also builds a strong foundation for advanced mathematical concepts, such as calculus and linear algebra. Therefore, investing time and effort in mastering factoring techniques is a worthwhile endeavor for anyone pursuing a career in mathematics or related fields.

By following these steps and practicing regularly, you can become proficient in factoring quadratic expressions. Factoring is a crucial skill in algebra and will help you in solving more complex problems in the future.

Choices:

$2x$
$8$
$x^2$
$4$
$4x$
$3$
$1$