Solving 2x^2 + 5x + 7 = 4x - 6 Find A, B, C Coefficients
Finding the values of variables that satisfy an equation is a fundamental skill in algebra. In this article, we will explore how to solve a quadratic equation and determine the coefficients a, b, and c in the standard form. Specifically, we will focus on the equation 2x^2 + 5x + 7 = 4x - 6. This equation, at first glance, might seem straightforward, but it requires careful manipulation to bring it into the standard quadratic form, which is essential for identifying the coefficients. Understanding how to rearrange and simplify equations like this is crucial for tackling more complex algebraic problems and is a building block for further studies in mathematics and related fields. The process involves combining like terms, setting the equation equal to zero, and then recognizing the coefficients that correspond to the quadratic, linear, and constant terms. This article will provide a step-by-step guide to navigate this process, ensuring a clear understanding of how to solve quadratic equations and extract the necessary information.
Understanding Quadratic Equations
Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is generally expressed as ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. The condition a ≠0 is essential because if a were zero, the equation would reduce to a linear equation. The values of a, b, and c play a crucial role in determining the nature and number of solutions (or roots) of the quadratic equation. For instance, the discriminant (b^2 - 4ac) of the quadratic equation provides valuable information about the roots: if the discriminant is positive, there are two distinct real roots; if it's zero, there is exactly one real root (a repeated root); and if it's negative, there are no real roots, but rather two complex conjugate roots. Mastering the identification and interpretation of these coefficients is a fundamental step in solving quadratic equations and understanding their properties. Furthermore, recognizing the standard form allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each method has its own advantages and is suited to different types of quadratic equations, making the understanding of a, b, and c even more critical in choosing the most efficient solution strategy.
Step-by-Step Solution for 2x^2 + 5x + 7 = 4x - 6
To find the values of a, b, and c for the given equation, 2x^2 + 5x + 7 = 4x - 6, we need to follow a systematic approach. The first crucial step is to rearrange the equation into the standard quadratic form ax^2 + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. To achieve this, we begin by subtracting 4x from both sides of the equation. This operation ensures that we maintain the equality while moving the linear term to the left-hand side. The equation then becomes 2x^2 + 5x - 4x + 7 = -6. Next, we simplify the equation by combining the like terms. In this case, we combine the x terms (5x and -4x) to get x. The equation is now 2x^2 + x + 7 = -6. The following step involves moving the constant term to the left side. We add 6 to both sides of the equation to eliminate the -6 on the right side. This gives us 2x^2 + x + 7 + 6 = 0. Finally, we simplify the constant terms (7 and 6) by adding them together. This results in the equation 2x^2 + x + 13 = 0. Now, the equation is in the standard quadratic form, allowing us to easily identify the coefficients a, b, and c. This meticulous step-by-step transformation ensures accuracy and provides a clear path to identifying the coefficients, which is a foundational skill in algebra.
Step 1: Rearrange the Equation
Our initial equation is 2x^2 + 5x + 7 = 4x - 6. The primary goal is to bring this equation into the standard quadratic form, ax^2 + bx + c = 0. This form is essential because it allows us to readily identify the coefficients a, b, and c, which are crucial for solving the equation using methods like the quadratic formula or factoring. The first step in rearranging the equation involves moving all terms to one side, leaving zero on the other side. This is achieved by performing algebraic operations that maintain the equality of the equation. We start by subtracting 4x from both sides of the equation. This action eliminates the linear term on the right side and moves it to the left side, where it can be combined with the existing x term. The equation now looks like this: 2x^2 + 5x + 7 - 4x = 4x - 6 - 4x, which simplifies to 2x^2 + 5x - 4x + 7 = -6. This step is a crucial part of the process, as it begins the transformation of the original equation into the standard form, setting the stage for the subsequent steps. The careful manipulation of terms while maintaining equality is a fundamental principle in algebra and a key skill for solving equations of any degree. By strategically moving terms, we are one step closer to identifying the coefficients a, b, and c, which will ultimately help us understand and solve the quadratic equation.
Step 2: Simplify the Equation
After rearranging the equation to 2x^2 + 5x - 4x + 7 = -6, the next crucial step is to simplify it by combining like terms. This process involves identifying terms that have the same variable and exponent and then performing the indicated arithmetic operations. In this case, we focus on the terms involving x: 5x and -4x. Combining these terms means subtracting 4x from 5x, which results in x. The equation now becomes 2x^2 + x + 7 = -6. This simplification step is significant because it reduces the number of terms in the equation, making it easier to manage and analyze. By combining like terms, we not only make the equation more concise but also bring it closer to the standard quadratic form ax^2 + bx + c = 0. Simplifying the equation is a fundamental skill in algebra and is essential for solving various types of equations. It involves careful attention to detail and a solid understanding of algebraic operations. By reducing the complexity of the equation, we can more easily identify the coefficients a, b, and c and proceed with solving the quadratic equation. This step showcases the importance of precision and organization in mathematical problem-solving, ensuring that we are on the right track to finding the solution.
Step 3: Move the Constant Term
Having simplified the equation to 2x^2 + x + 7 = -6, the next step in achieving the standard quadratic form is to move the constant term from the right side to the left side. The goal is to have all terms on one side of the equation, leaving zero on the other side, which aligns with the standard form ax^2 + bx + c = 0. To move the constant term, which is -6 in this case, we perform the inverse operation, which is adding 6 to both sides of the equation. This ensures that the equality of the equation is maintained. Adding 6 to both sides gives us 2x^2 + x + 7 + 6 = -6 + 6, which simplifies to 2x^2 + x + 7 + 6 = 0. This step is critical because it isolates all the terms of the quadratic equation on one side, paving the way for the final simplification and identification of the coefficients. The process of moving the constant term demonstrates the importance of using inverse operations to manipulate equations while preserving their balance. This skill is a fundamental aspect of algebra and is essential for solving a wide range of equations. By completing this step, we are now very close to expressing the equation in its standard form, which will allow us to easily determine the values of a, b, and c.
Step 4: Final Simplification
After moving the constant term, our equation stands as 2x^2 + x + 7 + 6 = 0. The final simplification step involves combining the constant terms on the left side of the equation. We have two constant terms: 7 and 6. Adding these together, we get 7 + 6 = 13. Therefore, the equation simplifies to 2x^2 + x + 13 = 0. This simplified form is the standard quadratic form, ax^2 + bx + c = 0, which is the ultimate goal of the rearrangement process. By reaching this form, we have successfully transformed the original equation into a format that allows us to easily identify the coefficients a, b, and c. This step highlights the importance of thoroughness and attention to detail in algebraic manipulations. By ensuring that all like terms are combined, we arrive at the most concise and manageable form of the equation. The final simplification is not just a cosmetic step; it is a crucial part of the solution process, as it directly leads to the identification of the coefficients, which are essential for solving the quadratic equation using methods like the quadratic formula or factoring. With the equation now in standard form, we are well-prepared to determine the values of a, b, and c and proceed with solving for the variable x if required.
Identifying a, b, and c
Now that we have successfully transformed the equation 2x^2 + 5x + 7 = 4x - 6 into the standard quadratic form 2x^2 + x + 13 = 0, we can easily identify the coefficients a, b, and c. Recall that the standard form of a quadratic equation is ax^2 + bx + c = 0, where a is the coefficient of the quadratic term (x^2), b is the coefficient of the linear term (x), and c is the constant term. By comparing our simplified equation 2x^2 + x + 13 = 0 with the standard form, we can directly read off the values of a, b, and c. The coefficient of the x^2 term is 2, so a = 2. The coefficient of the x term is 1 (since x is the same as 1x), so b = 1. The constant term is 13, so c = 13. Therefore, we have successfully identified the coefficients: a = 2, b = 1, and c = 13. This process of identifying coefficients is a fundamental skill in algebra and is crucial for solving quadratic equations using various methods, such as the quadratic formula, completing the square, or factoring. The ability to accurately determine the values of a, b, and c is essential for applying these methods correctly and finding the solutions (or roots) of the quadratic equation. Furthermore, the values of a, b, and c provide valuable information about the nature of the quadratic equation, such as the shape and position of its graph (a parabola) and the existence and type of its roots.
Significance of a, b, and c
The coefficients a, b, and c in a quadratic equation ax^2 + bx + c = 0 are not just arbitrary numbers; they hold significant mathematical meaning and influence the behavior and solutions of the equation. The coefficient a, in particular, plays a crucial role. It determines the shape and direction of the parabola that represents the quadratic equation when graphed. If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. The magnitude of a also affects the “width” of the parabola; a larger absolute value of a results in a narrower parabola, while a smaller absolute value results in a wider parabola. The coefficient b is related to the axis of symmetry of the parabola, which is a vertical line that passes through the vertex (the highest or lowest point) of the parabola. The x-coordinate of the vertex is given by -b/(2a), highlighting the influence of b on the parabola's position in the coordinate plane. The constant term c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. This is because when x = 0, the equation simplifies to y = c. Beyond the graphical interpretation, a, b, and c are essential for calculating the discriminant (b^2 - 4ac), which provides critical information about the nature of the roots of the quadratic equation. As mentioned earlier, the discriminant tells us whether the equation has two distinct real roots, one real root (a repeated root), or no real roots (two complex conjugate roots). In summary, the coefficients a, b, and c are fundamental parameters that govern the characteristics and solutions of quadratic equations, making their understanding and identification crucial for anyone studying algebra and related fields.
Conclusion
In this article, we have thoroughly explored the process of finding the values of a, b, and c for the equation 2x^2 + 5x + 7 = 4x - 6. We began by understanding the standard form of a quadratic equation, ax^2 + bx + c = 0, and the significance of the coefficients a, b, and c. We then embarked on a step-by-step journey to rearrange the given equation into the standard form. This involved several key steps: first, rearranging the equation by moving all terms to one side; second, simplifying the equation by combining like terms; third, moving the constant term to the left side; and finally, performing a final simplification to achieve the standard form. Through these careful manipulations, we transformed the original equation into 2x^2 + x + 13 = 0. With the equation in standard form, we were able to easily identify the coefficients: a = 2, b = 1, and c = 13. We also discussed the significance of these coefficients, highlighting their influence on the shape and position of the parabola representing the quadratic equation, as well as their role in determining the nature of the roots. This process not only demonstrates the importance of algebraic manipulation skills but also underscores the fundamental role of quadratic equations in mathematics and various applications. Mastering the identification of a, b, and c is a crucial step in solving quadratic equations and understanding their properties, laying a strong foundation for further studies in algebra and related fields.
