Solving The Quadratic Equation 4x² + 3x = 24 - X
In the realm of mathematics, quadratic equations hold a prominent position, appearing in various scientific and engineering applications. These equations, characterized by their second-degree polynomial form, often present a fascinating challenge to solve. This article delves into the intricacies of solving a specific quadratic equation: 4x² + 3x = 24 - x. We will embark on a step-by-step journey, employing algebraic techniques to unravel the roots of this equation. This article will provide a detailed explanation of the solution process, ensuring clarity and comprehension for readers of all backgrounds. Understanding how to solve quadratic equations is a fundamental skill in algebra, and this guide aims to equip you with the necessary knowledge and confidence to tackle similar problems.
Understanding Quadratic Equations
Before we dive into the solution, let's briefly revisit the essence of quadratic equations. A quadratic equation is defined as an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to a quadratic equation, often called roots or zeros, are the values of 'x' that satisfy the equation. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. There are several methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. The choice of method often depends on the specific equation and the solver's preference. In this article, we will primarily focus on using algebraic manipulation and the quadratic formula to determine the solutions to our target equation. Understanding the nature of quadratic equations and their various solution methods is crucial for success in algebra and related fields. The ability to solve these equations unlocks a wide range of problem-solving capabilities.
Transforming the Equation
The initial step in solving 4x² + 3x = 24 - x is to rearrange the equation into the standard quadratic form, ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. To achieve this, we add 'x' to both sides and subtract '24' from both sides. This process yields the following transformation:
4x² + 3x + x - 24 = 0
Combining the 'x' terms, we simplify the equation further:
4x² + 4x - 24 = 0
Now, we have the equation in the standard quadratic form. This form is essential for applying various solution methods, including the quadratic formula. The coefficients 'a', 'b', and 'c' are now clearly identifiable as 4, 4, and -24, respectively. Recognizing these coefficients is a critical step in utilizing the quadratic formula. By transforming the original equation into standard form, we set the stage for a systematic solution process. This algebraic manipulation is a fundamental technique in solving various types of equations, not just quadratic equations. The ability to rearrange and simplify equations is a cornerstone of algebraic proficiency.
Simplifying the Equation (Optional)
Before directly applying the quadratic formula, we can simplify the equation 4x² + 4x - 24 = 0 by dividing all terms by the greatest common divisor of the coefficients, which is 4. This simplification can make the subsequent calculations easier. Dividing both sides of the equation by 4, we get:
x² + x - 6 = 0
This simplified equation is equivalent to the original equation but has smaller coefficients. The coefficients are now a = 1, b = 1, and c = -6. While this simplification is optional, it often reduces the complexity of the quadratic formula calculations. The goal is to work with smaller numbers, which minimizes the chance of arithmetic errors. This technique of simplifying equations by dividing by a common factor is a valuable tool in algebraic problem-solving. It demonstrates a practical approach to making calculations more manageable. However, it's important to remember that this step is not strictly necessary and the quadratic formula can be applied directly to the original equation as well.
Applying the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. It provides a direct method for finding the roots, regardless of whether the equation can be easily factored. The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
In our simplified equation, x² + x - 6 = 0, we have a = 1, b = 1, and c = -6. Substituting these values into the quadratic formula, we get:
x = (-1 ± √(1² - 4 * 1 * -6)) / (2 * 1)
Simplifying the expression under the square root:
x = (-1 ± √(1 + 24)) / 2
x = (-1 ± √25) / 2
x = (-1 ± 5) / 2
Now we have two possible solutions, one with the plus sign and one with the minus sign. The quadratic formula is a fundamental concept in algebra, and mastering its application is essential for solving a wide range of problems. This formula guarantees a solution for any quadratic equation, even those that cannot be factored easily. Understanding the derivation and application of the quadratic formula is a key step in developing algebraic fluency. The formula provides a systematic approach to finding the roots of a quadratic equation, eliminating the need for trial and error.
Calculating the Roots
Let's calculate the two roots using the quadratic formula result, x = (-1 ± 5) / 2. First, we consider the positive case:
x₁ = (-1 + 5) / 2
x₁ = 4 / 2
x₁ = 2
So, one root is 2. Now, we consider the negative case:
x₂ = (-1 - 5) / 2
x₂ = -6 / 2
x₂ = -3
Therefore, the two roots of the equation x² + x - 6 = 0 (and equivalently, 4x² + 4x - 24 = 0 and 4x² + 3x = 24 - x) are 2 and -3. These are the values of 'x' that satisfy the original equation. Finding these roots is the ultimate goal of solving the quadratic equation. The ability to accurately calculate these roots demonstrates a strong understanding of algebraic manipulation and the application of the quadratic formula. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis. Understanding the graphical representation of these roots provides a deeper insight into the nature of quadratic equations.
Verifying the Solutions
To ensure the accuracy of our solutions, it's always a good practice to verify them by substituting them back into the original equation, 4x² + 3x = 24 - x. Let's first verify the solution x = 2:
4(2)² + 3(2) = 24 - 2
4(4) + 6 = 22
16 + 6 = 22
22 = 22
The equation holds true for x = 2. Now, let's verify the solution x = -3:
4(-3)² + 3(-3) = 24 - (-3)
4(9) - 9 = 24 + 3
36 - 9 = 27
27 = 27
The equation also holds true for x = -3. Since both solutions satisfy the original equation, we can confidently conclude that our solutions are correct. This verification step is crucial in preventing errors and ensuring the accuracy of our work. By substituting the solutions back into the original equation, we confirm that they are indeed the roots of the equation. This process reinforces the understanding of what it means to solve an equation and provides a sense of confidence in the results obtained.
Conclusion
In this comprehensive exploration, we successfully navigated the solution process for the quadratic equation 4x² + 3x = 24 - x. We began by transforming the equation into the standard quadratic form, then optionally simplified it by dividing by a common factor. We then applied the quadratic formula, a powerful tool for solving quadratic equations, to determine the roots. Finally, we meticulously verified our solutions by substituting them back into the original equation. The solutions we found are x = 2 and x = -3, corresponding to option C. This journey exemplifies the systematic approach required to solve quadratic equations and reinforces the importance of algebraic manipulation and the application of the quadratic formula. Understanding quadratic equations is a fundamental skill in mathematics, and this article has provided a detailed guide to mastering this skill. The ability to solve these equations opens doors to a wide range of applications in various fields, including physics, engineering, and economics. This exploration has not only provided the solution to a specific equation but also equipped readers with the knowledge and confidence to tackle similar problems in the future. The process of solving quadratic equations is a valuable exercise in developing problem-solving skills and analytical thinking.
