Factoring Quadratic Expressions A Step By Step Solution

by ADMIN 56 views

In the realm of mathematics, particularly algebra, factoring expressions plays a crucial role in simplifying complex equations and solving problems. This article delves into the process of factoring a given quadratic expression, providing a step-by-step guide to arrive at the correct factored form. We will explore the underlying principles, demonstrate the application of these principles to the given expression, and elucidate why one particular option stands out as the accurate factored form. Join us on this mathematical journey as we unravel the intricacies of factoring and empower you to tackle similar problems with confidence.

Understanding the Fundamentals of Factoring

At its core, factoring is the process of decomposing a mathematical expression into a product of simpler expressions. In the context of quadratic expressions, which are expressions of the form ax^2 + bx + c, factoring involves finding two binomials (expressions with two terms) that, when multiplied together, yield the original quadratic expression. This process is akin to reverse multiplication, where we start with the result and work backward to find the factors that produced it.

Factoring quadratic expressions is an essential skill in algebra for several reasons. First and foremost, it simplifies expressions, making them easier to manipulate and solve. Factored expressions often reveal hidden structures and relationships within the original expression, providing valuable insights. Moreover, factoring is instrumental in solving quadratic equations, which are equations of the form ax^2 + bx + c = 0. By factoring the quadratic expression, we can find the values of x that make the equation true, known as the roots or solutions of the equation.

There are several techniques for factoring quadratic expressions, each suited to different types of expressions. One common technique is to look for common factors among the terms of the expression. For example, in the expression 2x^2 + 4x, both terms have a common factor of 2x, which can be factored out to obtain 2x(x + 2). Another technique is to use the difference of squares pattern, which states that a^2 - b^2 can be factored as (a + b)(a - b). This pattern is particularly useful when dealing with expressions that involve the difference of two perfect squares.

For more complex quadratic expressions, we often employ the method of factoring by grouping. This method involves rearranging the terms of the expression and grouping them in pairs, then factoring out the greatest common factor from each pair. If the resulting expressions have a common binomial factor, we can factor it out to obtain the factored form of the original expression. Additionally, we can use the quadratic formula to find the roots of the expression and then work backwards to find the factored form. No matter the technique, the key to successful factoring is to carefully analyze the expression, identify any patterns or common factors, and apply the appropriate method systematically.

Deconstructing the Given Expression: A Step-by-Step Approach

The expression we aim to factor is: $4(x^2 - 2x) - 2(x^2 - 3)$.

To embark on this factoring journey, we must first simplify the expression by distributing the constants and combining like terms. This initial step lays the foundation for subsequent factoring techniques.

  1. Distribute the constants:

    • Multiply 4 by each term inside the first parenthesis: 4 * x^2 = 4x^2 and 4 * -2x = -8x.
    • Multiply -2 by each term inside the second parenthesis: -2 * x^2 = -2x^2 and -2 * -3 = 6.
    • This yields the expanded expression: 4x^2 - 8x - 2x^2 + 6.

  2. Combine like terms:

    • Identify terms with the same variable and exponent: 4x^2 and -2x^2 are like terms, and -8x and 6 are constant terms.
    • Combine the coefficients of like terms: 4x^2 - 2x^2 = 2x^2, and the other terms remain as -8x + 6.
    • The simplified expression is now: 2x^2 - 8x + 6.

Having simplified the expression, we can now proceed to factor it. The next step involves identifying common factors among the terms. In this case, we observe that all three terms (2x^2, -8x, and 6) share a common factor of 2. Factoring out this common factor simplifies the expression further and makes it easier to factor the remaining quadratic expression.

  1. Factor out the greatest common factor (GCF):
    • The GCF of 2x^2, -8x, and 6 is 2.
    • Divide each term by 2 and write the expression as a product: 2(x^2 - 4x + 3).

Now, we focus on factoring the quadratic expression inside the parenthesis: x^2 - 4x + 3. This quadratic expression is in the standard form ax^2 + bx + c, where a = 1, b = -4, and c = 3. To factor this expression, we need to find two numbers that multiply to give c (3) and add up to b (-4). These numbers will be the constants in the factored form of the quadratic expression.

Identifying the Correct Factorization: A Process of Elimination

To find the two numbers that multiply to 3 and add up to -4, we can list the factors of 3: 1 and 3. Since we need the numbers to add up to a negative number (-4), we consider the negative factors of 3: -1 and -3. Indeed, -1 multiplied by -3 equals 3, and -1 plus -3 equals -4. These are the numbers we need.

  1. Factor the quadratic expression:

    • Using the numbers -1 and -3, we can write the quadratic expression as a product of two binomials: (x - 1)(x - 3).

  2. Write the complete factored form:

    • Don't forget the GCF we factored out earlier. The complete factored form of the original expression is: 2(x - 1)(x - 3).

Now, let's examine the given options and determine which one matches our factored form:

A. 2(x + 1)(x + 3) B. (2x + 3)(x + 1) C. 2(x - 1)(x - 3) D. (2x - 3)(x - 1)

By comparing our factored form, 2(x - 1)(x - 3), with the options, we can clearly see that option C, 2(x - 1)(x - 3), is the correct answer. The other options either have incorrect signs or do not have the correct factors.

This methodical approach, involving simplification, factoring out the GCF, and factoring the quadratic expression, allows us to confidently identify the correct factored form. It's a testament to the power of breaking down complex problems into manageable steps.

Why Option C Reigns Supreme: A Detailed Explanation

Option C, 2(x - 1)(x - 3), stands as the undisputed champion for several compelling reasons. It perfectly encapsulates the factored form of the given expression, derived through a rigorous and systematic approach. Let's delve into the specific reasons why this option triumphs over the others.

First and foremost, option C accurately reflects the factored form obtained through our step-by-step process. We meticulously simplified the original expression, factored out the greatest common factor (GCF), and then factored the resulting quadratic expression. The final result, 2(x - 1)(x - 3), directly corresponds to option C. This alignment between our derived factored form and option C solidifies its validity.

Furthermore, option C adheres to the fundamental principles of factoring. Factoring is the process of expressing an expression as a product of simpler expressions. Option C embodies this principle by presenting the original expression as a product of a constant (2) and two binomials (x - 1) and (x - 3). This decomposition into simpler factors is the essence of factoring, and option C flawlessly demonstrates this concept.

In contrast, the other options falter in their representation of the factored form. Options A and B, 2(x + 1)(x + 3) and (2x + 3)(x + 1) respectively, contain incorrect signs within the binomials. These options would expand to a different quadratic expression than the one we started with, indicating they are not the correct factored form. Option D, (2x - 3)(x - 1), while having the correct signs, does not expand to the original expression either. This discrepancy further highlights the accuracy of option C.

To further validate option C, we can expand the factored form and verify that it matches the simplified expression. Expanding 2(x - 1)(x - 3), we get:

2(x^2 - 3x - x + 3) = 2(x^2 - 4x + 3) = 2x^2 - 8x + 6

This expanded form perfectly matches the simplified expression we obtained earlier, reinforcing the correctness of option C. This verification step provides an extra layer of confidence in our answer.

In conclusion, option C, 2(x - 1)(x - 3), reigns supreme as the correct factored form due to its direct correspondence with our derived factored form, its adherence to the principles of factoring, and its successful expansion back to the simplified expression. This option embodies the essence of factoring and provides the most accurate representation of the given expression in its factored form.

Concluding Thoughts: Mastering the Art of Factoring

Factoring quadratic expressions is an indispensable skill in the arsenal of any mathematics enthusiast. It empowers us to simplify complex equations, solve problems with elegance, and gain deeper insights into mathematical relationships. In this article, we embarked on a journey to unravel the factored form of a specific quadratic expression, but the principles and techniques we explored extend far beyond this single example.

The key takeaways from this exploration are the importance of systematic simplification, the power of identifying common factors, and the strategic application of factoring techniques. By breaking down complex expressions into manageable steps, we can conquer even the most challenging factoring problems. Remember to always look for opportunities to simplify, factor out the GCF, and then apply appropriate factoring methods based on the structure of the expression.

Furthermore, practice is paramount in mastering the art of factoring. The more you practice, the more comfortable you will become with identifying patterns, applying different techniques, and verifying your answers. Challenge yourself with a variety of quadratic expressions, and don't be afraid to make mistakes – they are valuable learning opportunities.

Factoring is not just a mechanical process; it's an exercise in problem-solving, critical thinking, and mathematical intuition. As you hone your factoring skills, you will develop a deeper appreciation for the beauty and elegance of mathematics. So, embrace the challenge, delve into the world of factoring, and unlock the hidden structures within mathematical expressions.

In conclusion, we have successfully navigated the intricacies of factoring the given quadratic expression, arriving at the correct factored form: 2(x - 1)(x - 3). This journey has not only provided us with a specific answer but has also equipped us with the tools and understanding to tackle similar problems with confidence. Embrace the power of factoring, and let it illuminate your path in the fascinating world of mathematics.