Solving X² - 14x + 31 = 63 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, characterized by a variable raised to the power of two, frequently appear in various mathematical and scientific contexts. This article will delve into the process of solving the quadratic equation x² - 14x + 31 = 63. We will explore different methods, providing a comprehensive guide for readers to understand and apply these techniques effectively. Understanding how to manipulate and solve quadratic equations opens doors to advanced mathematical concepts and practical applications.
The ability to solve quadratic equations is crucial in diverse fields. In physics, these equations are used to model projectile motion and other phenomena involving acceleration. In engineering, they are essential for designing structures, analyzing circuits, and optimizing processes. Even in economics and finance, quadratic equations can be used to model cost functions and profit maximization scenarios. By mastering the techniques presented in this article, readers will gain a valuable tool for problem-solving in various disciplines. The techniques discussed here include algebraic manipulation, factoring, completing the square, and applying the quadratic formula. Each method offers a unique approach, allowing for flexibility in tackling different types of quadratic equations. Understanding these methods not only helps in solving equations but also enhances overall mathematical reasoning and analytical skills.
Furthermore, the process of solving quadratic equations reinforces important mathematical concepts such as factoring, algebraic manipulation, and the properties of real numbers. These concepts are foundational for higher-level mathematics, including calculus and differential equations. Therefore, dedicating time to mastering quadratic equations is an investment in a broader mathematical education. In the following sections, we will break down the solution process into manageable steps, providing clear explanations and examples along the way. Whether you are a student learning algebra or someone seeking to refresh your math skills, this guide aims to provide a thorough understanding of how to solve quadratic equations.
Understanding Quadratic Equations
Before diving into the solution, it's crucial to understand the standard form of a quadratic equation: ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Recognizing this form is the first step in solving any quadratic equation. In our given equation, x² - 14x + 31 = 63, we need to rearrange it to fit this standard form. This involves subtracting 63 from both sides of the equation to set it equal to zero.
Quadratic equations arise naturally in many real-world applications, making their understanding essential. For instance, in physics, the trajectory of a projectile can be modeled using a quadratic equation, where the variable x represents time and the equation describes the height of the object. In engineering, these equations can be used to design parabolic reflectors, such as those used in satellite dishes and car headlights. The ability to solve quadratic equations is therefore not just an academic exercise but a practical skill with broad applications. Moreover, quadratic equations introduce the concept of complex numbers, which are essential in many areas of mathematics and engineering. The solutions to a quadratic equation can be real or complex, depending on the discriminant (b² - 4ac), which provides valuable information about the nature of the roots. Understanding the discriminant is crucial for predicting the type of solutions an equation will have, whether they are real and distinct, real and repeated, or complex conjugates.
The process of solving a quadratic equation also reinforces fundamental algebraic skills, such as factoring, completing the square, and applying the quadratic formula. Each method provides a different perspective on the equation and its solutions, fostering a deeper understanding of mathematical principles. Factoring, for example, involves breaking down the quadratic expression into simpler terms, which can then be easily solved. Completing the square is a powerful technique that transforms the equation into a form where the variable is isolated, revealing the solutions. And the quadratic formula provides a general solution that can be applied to any quadratic equation, regardless of its complexity. By mastering these methods, students develop a strong foundation in algebra and gain confidence in their problem-solving abilities.
Step 1: Rearranging the Equation
The first step is to rearrange the given equation, x² - 14x + 31 = 63, into the standard quadratic form. Subtract 63 from both sides to get: x² - 14x + 31 - 63 = 0, which simplifies to x² - 14x - 32 = 0. This form allows us to identify the coefficients a, b, and c, which are essential for the subsequent steps.
Rearranging the equation into standard form is a crucial step because it sets the stage for applying various solution methods. The standard form ax² + bx + c = 0 provides a clear structure that allows us to easily identify the coefficients a, b, and c, which are essential for factoring, completing the square, or using the quadratic formula. Without this initial rearrangement, it would be difficult to apply these methods effectively. Moreover, the standard form highlights the relationship between the coefficients and the roots of the equation, providing insights into the nature of the solutions.
For instance, the discriminant, which is calculated as 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 conjugate roots. Understanding the discriminant allows us to predict the type of solutions we should expect, which can guide our choice of solution method. Furthermore, the standard form facilitates the application of Vieta's formulas, which relate the sum and product of the roots to the coefficients of the equation. These formulas can be useful for checking the solutions and gaining a deeper understanding of the equation's properties. In the rearranged equation x² - 14x - 32 = 0, we can clearly identify that a = 1, b = -14, and c = -32. These coefficients will be used in the following steps to solve the equation using various methods, such as factoring and the quadratic formula. The ability to identify these coefficients accurately is a fundamental skill in solving quadratic equations and other algebraic problems.
Step 2: Solving by Factoring
Factoring involves expressing the quadratic expression as a product of two binomials. We look for two numbers that multiply to c (-32) and add up to b (-14). These numbers are -16 and 2. Thus, we can factor the equation as: (x - 16)(x + 2) = 0. Setting each factor to zero gives us the solutions: x - 16 = 0 or x + 2 = 0. Solving these equations yields x = 16 and x = -2.
Factoring is a powerful method for solving quadratic equations, particularly when the coefficients are integers and the roots are rational. The key to factoring is to find two numbers that satisfy specific conditions: their product must equal the constant term (c), and their sum must equal the coefficient of the linear term (b). In the equation x² - 14x - 32 = 0, we need to find two numbers that multiply to -32 and add up to -14. By systematically considering pairs of factors of -32, we can identify -16 and 2 as the desired numbers.
Once we have found these numbers, we can rewrite the quadratic expression as a product of two binomials: (x - 16)(x + 2). This factorization is based on the distributive property of multiplication over addition, which allows us to expand the product of the binomials back into the original quadratic expression. The next step is to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for x, leading to the solutions x - 16 = 0 and x + 2 = 0. Solving these linear equations gives us the roots of the quadratic equation: x = 16 and x = -2.
Factoring not only provides a method for finding solutions but also enhances our understanding of the relationship between the roots and the factors of the quadratic expression. Each root corresponds to a factor of the form (x - r), where r is the root. This connection is fundamental in algebra and is used in various applications, such as polynomial division and finding the roots of higher-degree polynomials. Furthermore, factoring reinforces our skills in algebraic manipulation and number theory, as we need to identify the factors of the constant term and their combinations to find the correct pair that satisfies the sum condition. In cases where factoring is not straightforward, other methods such as completing the square or the quadratic formula may be more appropriate, but factoring remains a valuable tool in our problem-solving toolkit.
Step 3: Verification
To ensure our solutions are correct, we substitute x = 16 and x = -2 back into the original equation: x² - 14x + 31 = 63.
For x = 16: (16)² - 14(16) + 31 = 256 - 224 + 31 = 63. This confirms that x = 16 is a solution.
For x = -2: (-2)² - 14(-2) + 31 = 4 + 28 + 31 = 63. This also confirms that x = -2 is a solution. Verification is a crucial step in the problem-solving process, especially in mathematics, where accuracy is paramount. It involves substituting the obtained solutions back into the original equation to ensure they satisfy the equation. This step helps to catch any errors that may have occurred during the solution process, such as mistakes in algebraic manipulation or incorrect application of formulas. In the case of quadratic equations, verification is particularly important because there are often two solutions, and both must be checked to ensure they are valid.
By substituting each solution back into the original equation, we are essentially testing whether the equation holds true for those values of x. If the equation is satisfied, then the solution is correct. If not, then an error has occurred, and we need to review our steps to identify and correct the mistake. Verification not only confirms the correctness of the solutions but also reinforces our understanding of the equation and its properties. It helps us to see how the solutions relate to the equation and to appreciate the symmetry and patterns that may exist.
In the context of the equation x² - 14x + 31 = 63, we found two potential solutions: x = 16 and x = -2. To verify these solutions, we substitute each value back into the original equation. For x = 16, we have (16)² - 14(16) + 31 = 256 - 224 + 31 = 63, which confirms that x = 16 is indeed a solution. Similarly, for x = -2, we have (-2)² - 14(-2) + 31 = 4 + 28 + 31 = 63, which confirms that x = -2 is also a solution. By performing these verifications, we can be confident that our solutions are accurate and that we have correctly solved the quadratic equation. This step not only provides a sense of accomplishment but also enhances our problem-solving skills and mathematical reasoning.
Alternative Method: Using the Quadratic Formula
For quadratic equations that are difficult to factor, the quadratic formula provides a general solution. The formula is: x = (-b ± √(b² - 4ac)) / (2a). For our equation, a = 1, b = -14, and c = -32. Substituting these values into the formula, we get:
x = (14 ± √((-14)² - 4(1)(-32))) / (2(1))
x = (14 ± √(196 + 128)) / 2
x = (14 ± √324) / 2
x = (14 ± 18) / 2
This gives us two solutions:
x = (14 + 18) / 2 = 32 / 2 = 16
x = (14 - 18) / 2 = -4 / 2 = -2
These solutions match those obtained by factoring.
The quadratic formula is a powerful tool for solving quadratic equations, particularly when factoring is not straightforward or when the roots are irrational or complex. It provides a general solution that can be applied to any quadratic equation in the standard form ax² + bx + c = 0. The formula is derived by completing the square on the general quadratic equation, and it expresses the roots in terms of the coefficients a, b, and c.
The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). The expression inside the square root, b² - 4ac, is known as the discriminant, and it plays a crucial role in determining the nature of 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 conjugate roots. Understanding the discriminant allows us to predict the type of solutions we should expect, which can be helpful in solving the equation.
In the case of our equation, x² - 14x - 32 = 0, we have a = 1, b = -14, and c = -32. Substituting these values into the quadratic formula, we get x = (14 ± √((-14)² - 4(1)(-32))) / (2(1)). Simplifying the expression under the square root, we have x = (14 ± √(196 + 128)) / 2, which further simplifies to x = (14 ± √324) / 2. Since the square root of 324 is 18, we have x = (14 ± 18) / 2. This gives us two solutions: x = (14 + 18) / 2 = 32 / 2 = 16 and x = (14 - 18) / 2 = -4 / 2 = -2. These solutions are the same as those obtained by factoring, which confirms the accuracy of both methods.
The quadratic formula not only provides a reliable method for solving quadratic equations but also enhances our understanding of the relationship between the coefficients and the roots. It demonstrates how the coefficients directly influence the solutions, and it highlights the importance of the discriminant in determining the nature of the roots. By mastering the quadratic formula, we gain a valuable tool for problem-solving in algebra and other areas of mathematics.
Conclusion
In conclusion, we have successfully solved the quadratic equation x² - 14x + 31 = 63 using both factoring and the quadratic formula. The solutions are x = 16 and x = -2. This exercise demonstrates the importance of understanding different methods for solving quadratic equations and verifying the solutions to ensure accuracy.
Solving quadratic equations is a fundamental skill in mathematics with applications in various fields, including physics, engineering, and economics. The ability to solve these equations allows us to model and analyze real-world problems involving parabolic relationships, such as the trajectory of a projectile or the design of a parabolic mirror. Mastering different methods for solving quadratic equations not only enhances our mathematical proficiency but also provides us with a versatile toolkit for problem-solving.
In this article, we explored two primary methods for solving the equation x² - 14x + 31 = 63: factoring and the quadratic formula. Factoring involves expressing the quadratic expression as a product of two binomials, which can then be set to zero to find the roots. This method is particularly effective when the coefficients are integers and the roots are rational. The quadratic formula, on the other hand, provides a general solution that can be applied to any quadratic equation, regardless of the nature of the roots. It is especially useful when factoring is not straightforward or when the roots are irrational or complex.
The process of solving the equation also involved rearranging it into the standard form ax² + bx + c = 0, which is a crucial step for both factoring and applying the quadratic formula. This rearrangement allows us to easily identify the coefficients a, b, and c, which are essential for the subsequent steps. Additionally, we emphasized the importance of verification, where we substituted the obtained solutions back into the original equation to ensure they satisfy the equation. This step helps to catch any errors that may have occurred during the solution process and reinforces our understanding of the equation and its properties.
By mastering these techniques, readers will gain confidence in their ability to solve quadratic equations and tackle more complex mathematical problems. The solutions x = 16 and x = -2 serve as a concrete example of how these methods can be applied to find the roots of a quadratic equation. This article provides a comprehensive guide for readers to understand and apply these techniques effectively, thereby enhancing their overall mathematical reasoning and analytical skills.
