Factoring Quadratic Expressions A Step-by-Step Guide For 3x^2 - 30x - 72

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In the realm of algebra, quadratic expressions hold a significant position, appearing in various mathematical contexts and real-world applications. Understanding how to factor these expressions is a fundamental skill, enabling us to solve equations, simplify expressions, and gain deeper insights into the behavior of quadratic functions. In this comprehensive guide, we will delve into the process of completely factoring the quadratic expression 3x2−30x−723x^2 - 30x - 72, providing a step-by-step approach that will equip you with the knowledge and confidence to tackle similar problems.

1. Understanding Quadratic Expressions

Before we embark on the factoring journey, it's crucial to have a solid grasp of what quadratic expressions are. A quadratic expression is a polynomial expression of the form ax2+bx+cax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The term 'ax2ax^2' is the quadratic term, 'bxbx' is the linear term, and 'c' is the constant term.

In our case, the expression 3x2−30x−723x^2 - 30x - 72 fits this form perfectly. Here, a=3a = 3, b=−30b = -30, and c=−72c = -72. Recognizing these coefficients is the first step toward unraveling the expression's factored form. Factoring a quadratic expression essentially means rewriting it as a product of two linear expressions. This process is like reverse engineering the distributive property, allowing us to break down complex expressions into simpler, manageable components.

Factoring quadratic expressions is not merely an algebraic exercise; it's a gateway to solving quadratic equations. When a quadratic expression is set equal to zero, factoring becomes a powerful technique for finding the roots or solutions of the equation. These roots represent the x-intercepts of the quadratic function's graph, providing valuable information about its behavior.

Moreover, factoring plays a crucial role in simplifying algebraic expressions and rational functions. By factoring both the numerator and denominator, common factors can be canceled out, leading to a more concise and manageable form. This simplification is essential in various mathematical contexts, including calculus and advanced algebra.

In the real world, quadratic expressions and their factored forms find applications in diverse fields, such as physics, engineering, and economics. For instance, projectile motion, the trajectory of a ball thrown in the air, can be modeled using quadratic equations. Factoring these equations allows us to determine the time it takes for the ball to reach its maximum height or the distance it travels before landing.

Understanding the significance of quadratic expressions and the power of factoring sets the stage for our exploration of the expression 3x2−30x−723x^2 - 30x - 72. By mastering the techniques involved in factoring, you'll not only enhance your algebraic skills but also gain a deeper appreciation for the interconnectedness of mathematical concepts.

2. Identifying the Greatest Common Factor (GCF)

The initial step in factoring any algebraic expression, including quadratics, is to identify and factor out the greatest common factor (GCF). The GCF is the largest factor that divides all the terms of the expression without leaving a remainder. This process simplifies the expression and often reveals underlying patterns that make further factoring easier.

In our expression, 3x2−30x−723x^2 - 30x - 72, we need to examine the coefficients and the variable terms to determine the GCF. Let's start with the coefficients: 3, -30, and -72. The largest number that divides all three of these coefficients is 3. Now, let's consider the variable terms: x2x^2, xx, and the constant term -72. The only common variable factor among these terms is 1, as the constant term does not have any variable.

Therefore, the GCF of the entire expression is 3. Factoring out the GCF involves dividing each term of the expression by 3 and writing the result in parentheses, with the GCF placed outside the parentheses:

3x2−30x−72=3(x2−10x−24)3x^2 - 30x - 72 = 3(x^2 - 10x - 24)

Notice how factoring out the GCF has significantly simplified the expression inside the parentheses. Instead of dealing with larger coefficients, we now have a quadratic expression with smaller, more manageable numbers. This simplification is a crucial step towards completely factoring the expression.

Factoring out the GCF is not just a matter of simplification; it's also a matter of accuracy. By removing the GCF, we ensure that the remaining expression is in its simplest form, making subsequent factoring steps more straightforward. Overlooking the GCF can lead to incorrect factoring or make the process more complicated than it needs to be.

Moreover, identifying the GCF provides valuable insights into the structure of the expression. It reveals the common building blocks that make up the expression, paving the way for further factorization. In our case, factoring out the GCF has transformed the original expression into a product of the GCF and a simpler quadratic expression, setting the stage for the next phase of factoring.

By mastering the art of identifying and factoring out the GCF, you'll not only simplify quadratic expressions but also develop a keen eye for recognizing patterns and structures in algebraic expressions. This skill is invaluable in various mathematical contexts, from solving equations to simplifying complex formulas.

3. Factoring the Simplified Quadratic Expression

After factoring out the GCF, we are left with the simplified quadratic expression x2−10x−24x^2 - 10x - 24. This expression is in the standard quadratic form ax2+bx+cax^2 + bx + c, where a=1a = 1, b=−10b = -10, and c=−24c = -24. Factoring this expression involves finding two binomials that, when multiplied together, yield the original quadratic.

One of the most common methods for factoring such expressions is the factoring by grouping method, also known as the ac method. This method involves finding two numbers that satisfy two conditions: their product equals the product of 'a' and 'c' (in this case, 1 * -24 = -24), and their sum equals 'b' (in this case, -10).

Let's systematically find these two numbers. We need two numbers that multiply to -24 and add up to -10. By considering the factors of -24, we can identify the pair -12 and 2. Indeed, -12 * 2 = -24 and -12 + 2 = -10. These are the numbers we need.

Now, we rewrite the middle term (-10x) using these two numbers:

x2−10x−24=x2−12x+2x−24x^2 - 10x - 24 = x^2 - 12x + 2x - 24

Next, we group the terms in pairs:

(x2−12x)+(2x−24)(x^2 - 12x) + (2x - 24)

From each pair, we factor out the GCF:

x(x−12)+2(x−12)x(x - 12) + 2(x - 12)

Notice that both terms now have a common binomial factor, (x−12)(x - 12). We factor out this common binomial:

(x−12)(x+2)(x - 12)(x + 2)

Thus, the factored form of the simplified quadratic expression x2−10x−24x^2 - 10x - 24 is (x−12)(x+2)(x - 12)(x + 2). This means that when we multiply these two binomials together, we get back the original quadratic expression.

Factoring by grouping is a powerful technique that can be applied to a wide range of quadratic expressions. It relies on the fundamental principles of factoring and the ability to identify patterns and relationships between the coefficients and terms of the expression. By mastering this method, you'll be well-equipped to factor complex quadratics with confidence.

Moreover, factoring quadratic expressions is not just a mathematical exercise; it's a skill that has practical applications in various fields. From solving equations to modeling real-world phenomena, factoring provides a powerful tool for understanding and manipulating quadratic relationships.

4. The Completely Factored Form

Having factored out the GCF and then factoring the simplified quadratic expression, we are now ready to present the completely factored form of the original expression, 3x2−30x−723x^2 - 30x - 72. Remember that we first factored out the GCF of 3, resulting in 3(x2−10x−24)3(x^2 - 10x - 24). Then, we factored the quadratic expression inside the parentheses as (x−12)(x+2)(x - 12)(x + 2).

To obtain the completely factored form, we simply combine these two results:

3x2−30x−72=3(x−12)(x+2)3x^2 - 30x - 72 = 3(x - 12)(x + 2)

This is the final answer, the completely factored form of the given quadratic expression. It represents the expression as a product of three factors: the constant 3, the binomial (x−12)(x - 12), and the binomial (x+2)(x + 2). Each factor is in its simplest form, and the expression cannot be factored further.

The completely factored form provides valuable insights into the expression's behavior and properties. For instance, it allows us to easily identify the roots of the quadratic equation 3x2−30x−72=03x^2 - 30x - 72 = 0. The roots are the values of 'x' that make the equation true, and they correspond to the values that make each factor equal to zero. In this case, the roots are x = 12 and x = -2.

The roots of a quadratic equation have a geometric interpretation as the x-intercepts of the parabola represented by the quadratic function. The factored form directly reveals these intercepts, making it a powerful tool for analyzing quadratic functions.

Moreover, the completely factored form can be used to simplify rational expressions, which are fractions where the numerator and denominator are polynomials. If the numerator or denominator contains a quadratic expression, factoring it can lead to the cancellation of common factors, resulting in a simplified expression. This simplification is crucial in various mathematical contexts, including calculus and advanced algebra.

In the real world, the completely factored form of quadratic expressions finds applications in diverse fields, such as physics, engineering, and economics. For instance, in projectile motion, the factored form of the quadratic equation that models the trajectory of a projectile can be used to determine the time it takes for the projectile to reach its maximum height or the distance it travels before landing.

By mastering the process of completely factoring quadratic expressions, you'll not only enhance your algebraic skills but also gain a deeper appreciation for the interconnectedness of mathematical concepts and their applications in the real world. The ability to factor expressions is a fundamental skill that will serve you well in various mathematical endeavors.

5. Conclusion: Mastering Factoring Quadratic Expressions

In this comprehensive guide, we have explored the process of completely factoring the quadratic expression 3x2−30x−723x^2 - 30x - 72. We began by understanding the nature of quadratic expressions and their significance in algebra and beyond. Then, we embarked on a step-by-step journey, first identifying and factoring out the greatest common factor (GCF), followed by factoring the simplified quadratic expression using the factoring by grouping method.

Finally, we combined the results to arrive at the completely factored form: 3(x−12)(x+2)3(x - 12)(x + 2). This form represents the expression as a product of three factors, each in its simplest form. We discussed how the completely factored form provides valuable insights into the expression's behavior, including its roots and their geometric interpretation as x-intercepts.

Factoring quadratic expressions is a fundamental skill in algebra, with applications in various mathematical contexts and real-world scenarios. It's a skill that requires a solid understanding of algebraic principles, attention to detail, and the ability to recognize patterns and relationships between the terms of the expression. By mastering the techniques involved in factoring, you'll not only enhance your algebraic skills but also gain a deeper appreciation for the interconnectedness of mathematical concepts.

Throughout this guide, we have emphasized the importance of each step in the factoring process. Identifying the GCF is crucial for simplifying the expression and making subsequent factoring steps easier. Factoring the simplified quadratic expression involves finding two binomials that, when multiplied together, yield the original quadratic. The factoring by grouping method provides a systematic approach to this process.

The completely factored form is the ultimate goal, representing the expression as a product of its simplest factors. This form provides valuable insights into the expression's behavior and properties, including its roots and their geometric interpretation.

Mastering factoring quadratic expressions is not just about memorizing steps; it's about developing a deep understanding of the underlying principles and applying them with confidence. With practice and perseverance, you can become proficient in factoring and unlock the power of this essential algebraic skill. So, embrace the challenge, explore the world of quadratic expressions, and embark on a journey of mathematical discovery!