Rewriting Quadratic Equations Understanding Standard Form Ax² + Bx + C = 0
In the realm of mathematics, particularly when dealing with quadratic equations, understanding the standard form is paramount. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. Transforming equations into this standard form allows for easier identification of coefficients and application of various problem-solving techniques, such as factoring, completing the square, or using the quadratic formula. This article aims to dissect the process of converting a given quadratic equation into its standard form, providing a step-by-step approach that will solidify your understanding and enhance your ability to manipulate quadratic equations effectively. We will take a specific example and walk through each stage of the transformation, ensuring clarity and precision in our explanation. By the end of this discussion, you will be well-equipped to tackle similar problems with confidence and accuracy. The journey of mastering quadratic equations begins with a firm grasp of the foundational concepts, and the standard form serves as the cornerstone for further exploration in this fascinating area of algebra.
The Importance of Standard Form
Before diving into the specifics of converting the equation, it's crucial to appreciate why the standard form ax² + bx + c = 0 is so vital. The standard form provides a consistent structure for analyzing quadratic equations. When an equation is in this form, it becomes straightforward to identify the coefficients a, b, and c, which are essential for using the quadratic formula, factoring, and completing the square. Each of these methods relies on these coefficients to find the roots or solutions of the equation. For instance, the quadratic formula, which is given by x = (-b ± √(b² - 4ac)) / (2a), directly uses the values of a, b, and c. Similarly, when factoring a quadratic equation, we often look for two numbers that multiply to c and add up to b, assuming a is equal to 1. Completing the square also requires a clear understanding of the coefficients to manipulate the equation effectively. Moreover, the standard form makes it easier to determine the nature of the roots of the quadratic equation. The discriminant, which is the part under the square root in the quadratic formula (b² - 4ac), provides valuable information about the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root (a repeated root); and if it is negative, the equation has two complex roots. Therefore, understanding the standard form is not just about rearranging terms; it's about unlocking a deeper understanding of the equation's properties and potential solutions. By having the equation in this format, mathematicians and students alike can quickly assess and solve a wide range of quadratic problems with greater efficiency and accuracy.
Converting the Equation x² + 8x + 10 = 3 into Standard Form
Let's consider the equation x² + 8x + 10 = 3. Our goal is to rewrite this equation in the standard form ax² + bx + c = 0. The key to achieving this is to manipulate the equation while maintaining its equality. The first step in this conversion process is to recognize that the standard form requires the equation to be set equal to zero. Currently, our equation is set equal to 3. To bring it into the desired form, we need to eliminate the 3 from the right side of the equation. This can be accomplished by subtracting 3 from both sides of the equation. Subtracting the same value from both sides ensures that the equation remains balanced, and the equality is preserved. When we subtract 3 from both sides, we get: x² + 8x + 10 - 3 = 3 - 3. This simplifies to: x² + 8x + 7 = 0. Now, let's examine the resulting equation. We have successfully rearranged the equation so that it is equal to zero. By doing this, we have taken a significant step towards achieving the standard form. The next step involves identifying the coefficients a, b, and c. In our transformed equation, x² + 8x + 7 = 0, we can see that a is the coefficient of the x² term, which is 1 (since x² is the same as 1x²). The coefficient b is the coefficient of the x term, which is 8. Finally, c is the constant term, which is 7. Now that we have identified a, b, and c, we can confidently state that the equation x² + 8x + 10 = 3, when written in standard form, becomes x² + 8x + 7 = 0. This transformation is a fundamental skill in algebra and serves as a foundation for solving more complex quadratic equations.
Identifying the Correct Option
Now that we've successfully converted the equation x² + 8x + 10 = 3 into its standard form, which is x² + 8x + 7 = 0, the next step is to identify the correct option among the given choices. The options presented are:
A. x² + 5x + 13 = 0 B. x² + 5x + 7 = 0 C. x² + 8x + 13 = 0 D. x² + 8x + 7 = 0
By carefully comparing our derived standard form equation, x² + 8x + 7 = 0, with the provided options, we can clearly see that option D, x² + 8x + 7 = 0, matches our result exactly. Options A, B, and C have different coefficients for the x term and the constant term, making them incorrect. Option A has a coefficient of 5 for the x term and a constant term of 13. Option B also has a coefficient of 5 for the x term but has a constant term of 7. Option C has the correct coefficient of 8 for the x term but has a constant term of 13. These discrepancies highlight the importance of accurate algebraic manipulation to arrive at the correct standard form. The process of converting and comparing underscores the precision required in mathematical problem-solving. Therefore, through our step-by-step conversion and careful comparison, we can confidently conclude that the correct option is D. This exercise demonstrates the significance of understanding the standard form of a quadratic equation and the methods for transforming equations into this form. Accurate conversion and comparison are essential skills for solving quadratic equations and for success in algebra in general.
Conclusion
In summary, we've successfully transformed the equation x² + 8x + 10 = 3 into its standard quadratic form, ax² + bx + c = 0, and identified the correct equivalent equation. The key steps involved subtracting 3 from both sides of the original equation to set it equal to zero, resulting in the standard form x² + 8x + 7 = 0. This process underscores the importance of understanding the properties of equations and how to manipulate them while maintaining equality. The standard form of a quadratic equation is crucial because it allows for easy identification of the coefficients a, b, and c, which are essential for solving the equation using methods like factoring, completing the square, or applying the quadratic formula. Furthermore, having the equation in standard form simplifies the analysis of the discriminant, which provides valuable information about the nature of the roots. By accurately converting the equation and carefully comparing it with the given options, we confidently selected option D, x² + 8x + 7 = 0, as the correct representation. This exercise not only reinforces the concept of standard form but also highlights the significance of precision and attention to detail in mathematical problem-solving. Mastering these fundamental skills is essential for tackling more complex algebraic problems and for success in higher-level mathematics. The ability to manipulate equations into standard forms is a powerful tool in the mathematical arsenal, enabling efficient and accurate solutions to a wide range of problems.
