Introduction
In this comprehensive article, we will delve into the intricate process of solving for x in the given equation: (6x² + 13x - 4) / (2x + 5) = (12x² + 5x - 2) / (4x + 3). This equation, a fascinating blend of rational expressions and quadratic forms, requires a strategic approach to unravel its solution. Mathematics, as a discipline, thrives on problem-solving, and this particular problem offers a rich opportunity to demonstrate various algebraic techniques. From factoring and cross-multiplication to identifying extraneous solutions, we will meticulously explore each step, ensuring a clear and thorough understanding. This journey through the equation will not only provide the solution for x but also reinforce fundamental mathematical principles, making it a valuable exercise for students, educators, and mathematics enthusiasts alike. So, let's embark on this mathematical expedition, armed with our knowledge and a determination to conquer the complexities of the equation.
Problem Statement
The problem at hand involves finding the values of x that satisfy the equation:
(6x² + 13x - 4) / (2x + 5) = (12x² + 5x - 2) / (4x + 3)
This equation presents a challenge due to its rational form, which necessitates careful manipulation to isolate x. The presence of quadratic expressions in both the numerator of each fraction adds another layer of complexity. To solve this equation effectively, we will employ a combination of algebraic techniques, including factoring, cross-multiplication, and the crucial step of checking for extraneous solutions. Extraneous solutions are values of x that emerge during the solving process but do not actually satisfy the original equation, often because they make the denominator of a fraction equal to zero. Therefore, our approach will be methodical, ensuring that each step is logically sound and that we account for all potential pitfalls. The ultimate goal is to find all valid values of x that make the equation true, providing a comprehensive solution to the problem.
Step-by-Step Solution
1. Factoring Quadratic Expressions
Our initial step in solving for x involves factoring the quadratic expressions in the numerators of both fractions. Factoring is a powerful technique that simplifies complex expressions by breaking them down into their constituent parts. Let's begin by factoring 6x² + 13x - 4. We are looking for two binomials that, when multiplied together, yield the original quadratic. By careful inspection and trial-and-error, we can factor this expression as follows:
6x² + 13x - 4 = (2x + 4)(3x - 1)
Next, we turn our attention to the second quadratic expression, 12x² + 5x - 2. Applying the same factoring principles, we seek two binomials that multiply to give this quadratic. After some consideration, we find that:
12x² + 5x - 2 = (3x + 2)(4x - 1)
By successfully factoring both quadratic expressions, we have transformed the original equation into a more manageable form. This step is crucial as it allows us to potentially simplify the equation further through cancellation or other algebraic manipulations. With the quadratics factored, we can now rewrite the equation as:
(2x + 4)(3x - 1) / (2x + 5) = (3x + 2)(4x - 1) / (4x + 3)
This factorization sets the stage for the next phase of our solution, where we will employ cross-multiplication to eliminate the fractions and move closer to isolating x.
2. Cross-Multiplication
With the quadratic expressions factored, the next strategic move is to employ cross-multiplication. This technique is invaluable when dealing with equations involving fractions, as it eliminates the denominators and transforms the equation into a more linear form. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa, setting these products equal to each other. Applying this to our equation:
(2x + 4)(3x - 1) / (2x + 5) = (3x + 2)(4x - 1) / (4x + 3)
We multiply (2x + 4)(3x - 1) by (4x + 3) and (3x + 2)(4x - 1) by (2x + 5). This yields the following equation:
(2x + 4)(3x - 1)(4x + 3) = (3x + 2)(4x - 1)(2x + 5)
This step effectively removes the fractions, but it also introduces polynomial expressions that need to be expanded and simplified. The resulting equation is a more complex polynomial equation, but it is free of fractions, making it easier to manipulate algebraically. The next phase involves expanding these products and simplifying the equation to consolidate terms and move towards isolating x.
3. Expanding and Simplifying
Following the cross-multiplication, we now face the task of expanding the products on both sides of the equation. This step is crucial for simplifying the equation and bringing like terms together. We start by expanding the left side of the equation:
(2x + 4)(3x - 1)(4x + 3)
First, we multiply (2x + 4) and (3x - 1):
(2x + 4)(3x - 1) = 6x² - 2x + 12x - 4 = 6x² + 10x - 4
Then, we multiply the result by (4x + 3):
(6x² + 10x - 4)(4x + 3) = 24x³ + 18x² + 40x² + 30x - 16x - 12 = 24x³ + 58x² + 14x - 12
Now, we expand the right side of the equation:
(3x + 2)(4x - 1)(2x + 5)
First, we multiply (3x + 2) and (4x - 1):
(3x + 2)(4x - 1) = 12x² - 3x + 8x - 2 = 12x² + 5x - 2
Then, we multiply the result by (2x + 5):
(12x² + 5x - 2)(2x + 5) = 24x³ + 60x² + 10x² + 25x - 4x - 10 = 24x³ + 70x² + 21x - 10
Now, we have the expanded equation:
24x³ + 58x² + 14x - 12 = 24x³ + 70x² + 21x - 10
This expanded form allows us to combine like terms and further simplify the equation. The next step involves moving all terms to one side of the equation and looking for ways to reduce it to a more manageable form, such as a quadratic or linear equation.
4. Consolidating Terms
With both sides of the equation expanded, the next step is to consolidate terms and simplify the equation. This process involves moving all terms to one side, ideally resulting in a polynomial equation set equal to zero. Starting from our expanded equation:
24x³ + 58x² + 14x - 12 = 24x³ + 70x² + 21x - 10
We subtract the right side of the equation from the left side to set the equation to zero:
(24x³ + 58x² + 14x - 12) - (24x³ + 70x² + 21x - 10) = 0
This simplifies to:
24x³ - 24x³ + 58x² - 70x² + 14x - 21x - 12 + 10 = 0
Combining like terms, we get:
-12x² - 7x - 2 = 0
To make the leading coefficient positive, we can multiply the entire equation by -1:
12x² + 7x + 2 = 0
We now have a quadratic equation in standard form. This simplified form is much easier to solve than the original equation. The next step involves solving this quadratic equation for x, which can be done through factoring, completing the square, or using the quadratic formula.
5. Solving the Quadratic Equation
Having consolidated and simplified the equation to a standard quadratic form, our focus now shifts to solving for x in the equation:
12x² + 7x + 2 = 0
There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, the quadratic formula is the most direct approach. The quadratic formula is given by:
x = [-b ± √(b² - 4ac)] / (2a)
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our equation, a = 12, b = 7, and c = 2. Plugging these values into the quadratic formula, we get:
x = [-7 ± √(7² - 4 * 12 * 2)] / (2 * 12)
Simplifying the expression under the square root:
7² - 4 * 12 * 2 = 49 - 96 = -47
So, the equation becomes:
x = [-7 ± √(-47)] / 24
Since the discriminant (the value under the square root) is negative, the solutions will be complex numbers. We can express the square root of -47 as √(-47) = i√47, where i is the imaginary unit (√-1). Thus, the solutions are:
x = [-7 ± i√47] / 24
This gives us two complex solutions for x:
x₁ = (-7 + i√47) / 24
x₂ = (-7 - i√47) / 24
These complex solutions are the values of x that satisfy the quadratic equation. However, it is crucial to check for extraneous solutions in the context of the original equation, especially since we dealt with rational expressions.
6. Checking for Extraneous Solutions
After finding the solutions for x, it is imperative to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation, often due to making a denominator zero. Our original equation was:
(6x² + 13x - 4) / (2x + 5) = (12x² + 5x - 2) / (4x + 3)
The denominators are (2x + 5) and (4x + 3). We need to ensure that our solutions do not make these denominators equal to zero. Let's set each denominator equal to zero and solve for x:
2x + 5 = 0
2x = -5
x = -5/2
4x + 3 = 0
4x = -3
x = -3/4
Thus, x = -5/2 and x = -3/4 are the values that would make the denominators zero. Our solutions, x₁ = (-7 + i√47) / 24 and x₂ = (-7 - i√47) / 24, are complex numbers. Since complex numbers are not equal to real numbers like -5/2 and -3/4, our solutions do not make the denominators zero. Therefore, neither solution is extraneous.
7. Final Answer
Having meticulously solved the equation and checked for extraneous solutions, we can now confidently state the final answers. The solutions for the equation
(6x² + 13x - 4) / (2x + 5) = (12x² + 5x - 2) / (4x + 3)
are the complex numbers:
x₁ = (-7 + i√47) / 24
x₂ = (-7 - i√47) / 24
These complex solutions are the only values of x that satisfy the original equation. Our journey through this problem has not only provided the solutions but also reinforced the importance of various algebraic techniques, such as factoring, cross-multiplication, and checking for extraneous solutions. The inclusion of complex numbers in the solution set adds an extra layer of mathematical sophistication to this problem.
Conclusion
In conclusion, solving the equation
(6x² + 13x - 4) / (2x + 5) = (12x² + 5x - 2) / (4x + 3)
has been a comprehensive exercise in algebraic manipulation and problem-solving. We navigated through the complexities of rational expressions and quadratic equations by employing a step-by-step approach. This began with factoring the quadratic expressions, which allowed us to simplify the equation and set the stage for further manipulation. Cross-multiplication was then used to eliminate the fractions, transforming the equation into a more manageable polynomial form. Expanding and simplifying the resulting expressions led us to a standard quadratic equation, which we solved using the quadratic formula. The solutions, complex numbers x₁ = (-7 + i√47) / 24 and x₂ = (-7 - i√47) / 24, were carefully checked for extraneous solutions to ensure their validity. This process highlighted the critical role of checking solutions, particularly in equations involving fractions. The final solutions underscore the importance of being comfortable with complex numbers in algebraic problem-solving. This detailed exploration not only provides the answer but also serves as a valuable educational resource, reinforcing key mathematical concepts and techniques. Solving such problems enhances one's mathematical acumen and problem-solving skills, making it a worthwhile endeavor for anyone interested in mathematics.