Solving 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0 A Comprehensive Guide

Introduction

In the realm of mathematics, quadratic equations hold a pivotal role, frequently encountered across various disciplines, ranging from physics and engineering to economics and computer science. A quadratic equation, characterized by its second-degree polynomial form, generally appears as ax^2 + bx + c = 0, where a, b, and c represent constants and x signifies the variable we aim to solve for. The equation presented, 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0, exemplifies a quadratic equation with coefficients that involve algebraic terms, thus presenting a more intricate challenge to solve. This comprehensive guide aims to dissect the methodologies for solving such an equation, offering a step-by-step approach to unravel the solutions for x. The techniques explored herein will not only empower you to tackle this specific problem but also equip you with the skills to navigate similar complex quadratic equations with confidence and precision. Understanding these methods is crucial for anyone delving into mathematical problem-solving, as quadratic equations form the bedrock of numerous advanced mathematical concepts and applications. This article will delve into the intricacies of factoring, the quadratic formula, and other relevant techniques, providing a robust toolkit for solving a wide array of quadratic equations. Our focus will remain steadfast on ensuring clarity and comprehension, making the journey through this mathematical landscape both enlightening and rewarding. The ability to solve quadratic equations efficiently is an invaluable asset, opening doors to further exploration in mathematics and related fields. So, let's embark on this journey together, unraveling the mysteries of quadratic equations and mastering the art of finding solutions.

Understanding the Quadratic Equation

Before diving into the solution, let's first understand the structure of the given quadratic equation: 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0. This equation is in the standard quadratic form ax^2 + bx + c = 0, where:

• a = 12ab
• b = -(9a^2 - 8b^2)
• c = -6ab

The coefficients 'a', 'b', and 'c' are algebraic expressions, making this problem more complex than a simple numerical quadratic equation. To solve for x, we can employ several methods, including factoring, completing the square, or using the quadratic formula. In this case, factoring appears to be a viable approach due to the structure of the coefficients. However, recognizing the appropriate factoring strategy requires a keen eye and a systematic approach. The process involves identifying two binomials that, when multiplied, yield the original quadratic expression. This often entails breaking down the middle term ('b' coefficient) into two parts that satisfy specific conditions related to the product and sum of the coefficients. Understanding the relationships between the coefficients and the roots of the quadratic equation is crucial for successful factoring. For instance, the product of the roots is related to 'c/a', and the sum of the roots is related to '-b/a'. By leveraging these relationships, we can strategically search for the appropriate factors. Additionally, the complexity introduced by the algebraic coefficients necessitates careful attention to detail and a methodical approach to avoid errors in the factoring process. Factoring, when applicable, often provides a more elegant and efficient solution compared to other methods, making it a valuable tool in the arsenal of any mathematician or problem solver.

Factoring the Quadratic Equation

To factor the quadratic equation 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0, we need to find two binomials that multiply to give us the original quadratic expression. This involves breaking down the middle term, -(9a^2 - 8b^2)x, into two terms such that their coefficients add up to -(9a^2 - 8b^2) and their product equals the product of the first and last terms (12ab * -6ab = -72a^2b2). The process of factoring quadratic equations, especially those with algebraic coefficients, demands a strategic and meticulous approach. It's akin to solving a puzzle, where the pieces are the terms of the quadratic expression, and the goal is to arrange them in a way that reveals the hidden binomial factors. This strategic decomposition of the middle term is crucial, as it lays the foundation for rewriting the quadratic expression in a form that can be easily factored by grouping. The challenge lies in identifying the correct pair of coefficients that satisfy the dual conditions of summing to the original middle term coefficient and multiplying to the product of the first and last term coefficients. This often requires a combination of intuition, pattern recognition, and systematic trial-and-error. Once the middle term is successfully decomposed, the next step involves grouping the terms in pairs and factoring out the greatest common factor from each pair. This process transforms the four-term expression into a product of two binomials, effectively revealing the factors of the quadratic equation. The ability to factor quadratic equations is not only a fundamental skill in algebra but also a gateway to solving more complex mathematical problems. It underscores the importance of understanding the underlying structure of mathematical expressions and the power of strategic manipulation to simplify and solve them.

Let's break down the middle term:

• We need two terms that add up to -(9a^2 - 8b^2) and multiply to -72a^2b2.
• Consider the terms -9a^2x and 8b^2x. These terms satisfy the conditions because:
  • -9a^2 + 8b^2 = -(9a^2 - 8b^2)
  • (-9a^2)(8b2) = -72a^2b2

Now, rewrite the equation:

12abx^2 - 9a^2x + 8b^2x - 6ab = 0

Next, factor by grouping:

3ax(4bx - 3a) + 2b(4bx - 3a) = 0

Now, we can factor out the common binomial (4bx - 3a):

(4bx - 3a)(3ax + 2b) = 0

Solving for x

Now that we have factored the quadratic equation into (4bx - 3a)(3ax + 2b) = 0, we can solve for x by setting each factor equal to zero:

1. 4bx - 3a = 0
   • 4bx = 3a
   • x = 3a / 4b
2. 3ax + 2b = 0
   • 3ax = -2b
   • x = -2b / 3a

Therefore, the solutions for x are 3a / 4b and -2b / 3a.

Verification of the Solutions

To ensure the accuracy of our solutions, it is imperative to substitute them back into the original quadratic equation: 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0. This verification process serves as a crucial step in confirming that the calculated values of x indeed satisfy the equation. By substituting each solution individually, we can meticulously check whether the left-hand side of the equation equals zero, thereby validating the correctness of our factoring and solving procedures. This step is particularly important when dealing with complex algebraic expressions, as it helps to identify and rectify any potential errors in the algebraic manipulation. The substitution process involves replacing every instance of 'x' in the original equation with the calculated solution and then simplifying the resulting expression. This simplification often requires careful application of algebraic rules and identities to ensure that all terms are correctly combined. If, after simplification, the left-hand side of the equation reduces to zero, it confirms that the substituted value of x is indeed a valid solution. This rigorous verification not only provides confidence in the solutions obtained but also enhances our understanding of the underlying algebraic relationships within the equation. It reinforces the importance of precision and attention to detail in mathematical problem-solving and highlights the value of double-checking our work to ensure accuracy.

Substituting x = 3a / 4b

12ab(3a / 4b)^2 - (9a^2 - 8b^2)(3a / 4b) - 6ab

12ab(9a^2 / 16b^2) - (27a^3 - 24ab^2) / 4b - 6ab

(27a^3 / 4b) - (27a^3 - 24ab^2) / 4b - 6ab

(27a^3 - 27a^3 + 24ab^2) / 4b - 6ab

(24ab^2 / 4b) - 6ab

6ab - 6ab = 0

Substituting x = -2b / 3a

12ab(-2b / 3a)^2 - (9a^2 - 8b^2)(-2b / 3a) - 6ab

12ab(4b^2 / 9a^2) + (18a^2b - 16b^3) / 3a - 6ab

(16b^3 / 3a) + (18a^2b - 16b^3) / 3a - 6ab

(16b^3 + 18a^2b - 16b^3) / 3a - 6ab

(18a^2b / 3a) - 6ab

6ab - 6ab = 0

Both solutions satisfy the equation.

Alternative Methods

While factoring provides an elegant solution in this case, it's worth noting that other methods, such as the quadratic formula, can also be used to solve this equation. The quadratic formula is a universal tool for solving quadratic equations of the form ax^2 + bx + c = 0 and is given by: x = [-b ± sqrt(b^2 - 4ac)] / 2a. Applying the quadratic formula to solve quadratic equations serves as a robust alternative, particularly when factoring proves challenging or impractical. This method's universality stems from its direct application to any quadratic equation expressed in the standard form ax^2 + bx + c = 0, where a, b, and c are coefficients. The formula elegantly encapsulates the relationship between these coefficients and the roots of the equation, providing a straightforward pathway to finding solutions. The process involves substituting the coefficients into the formula and simplifying the resulting expression, a task that may entail handling square roots and fractions. While the quadratic formula guarantees a solution, its application can sometimes be more computationally intensive than factoring, especially when dealing with complex coefficients or when the discriminant (b^2 - 4ac) is a perfect square. However, its reliability and broad applicability make it an indispensable tool in the arsenal of any mathematician or student. The formula not only provides solutions but also sheds light on the nature of the roots, revealing whether they are real or complex, rational or irrational, based on the discriminant's value. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (or a repeated root), and a negative discriminant indicates two complex roots. This comprehensive insight into the roots' characteristics underscores the quadratic formula's significance as a fundamental concept in algebra, offering both a method for solving equations and a tool for analyzing their solutions.

For our equation, a = 12ab, b = -(9a^2 - 8b^2), and c = -6ab. Plugging these values into the quadratic formula would yield the same solutions, although the process might be more involved.

Conclusion

In conclusion, we successfully solved the quadratic equation 12abx^2 - (9a^2 - 8b^2)x - 6ab = 0 by factoring. The solutions for x are 3a / 4b and -2b / 3a. We also verified these solutions by substituting them back into the original equation. While factoring was an efficient method for this particular problem, the quadratic formula provides a general approach for solving any quadratic equation. Mastering these techniques is crucial for tackling more advanced mathematical problems.