Solving 3x² - 18x + 5 = 47 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving quadratic equations is a fundamental skill. Quadratic equations, characterized by the highest power of the variable being 2, appear in various fields, from physics to engineering. This article dives deep into solving for x in the quadratic equation 3x² - 18x + 5 = 47. We'll explore the steps involved, the underlying principles, and the methods used to arrive at the solution. Our goal is not just to provide the answer, but to offer a comprehensive understanding of the process, ensuring you can tackle similar problems with confidence. This exploration will not only enhance your problem-solving abilities but also deepen your appreciation for the elegance and power of algebra. By the end of this guide, you'll be equipped with the knowledge and skills to confidently solve quadratic equations and apply these techniques in various mathematical contexts. The journey through this problem will also shed light on the versatility and importance of quadratic equations in real-world applications, making the learning experience both practical and enriching.
Understanding Quadratic Equations
Before diving into the solution, let's grasp the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of x that satisfy the equation. Quadratic equations can have two, one, or no real solutions, depending on the discriminant (b² - 4ac). Understanding the nature of these equations is crucial for solving them effectively. The coefficients a, b, and c play significant roles in determining the shape and position of the parabola represented by the quadratic equation when graphed. For instance, the sign of a determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). The roots of the equation correspond to the x-intercepts of the parabola. This graphical interpretation provides a visual understanding of the solutions. Moreover, quadratic equations are not just abstract mathematical concepts; they have wide-ranging applications in various fields, including physics (e.g., projectile motion), engineering (e.g., designing parabolic reflectors), and economics (e.g., modeling cost functions).
Setting up the Equation
Our equation is 3x² - 18x + 5 = 47. The first step is to set the equation to the standard form, which means making one side equal to zero. To do this, we subtract 47 from both sides of the equation:
3x² - 18x + 5 - 47 = 0
Simplifying, we get:
3x² - 18x - 42 = 0
Now, the equation is in the standard form, ax² + bx + c = 0, where a = 3, b = -18, and c = -42. This standard form is essential for applying various methods to solve the quadratic equation, such as factoring, completing the square, or using the quadratic formula. The act of rearranging the equation into standard form not only prepares it for these solution methods but also provides a clear view of the coefficients, which are crucial for subsequent steps. Recognizing and correctly identifying these coefficients is a fundamental skill in solving quadratic equations. Furthermore, this process highlights the importance of algebraic manipulation in mathematics, where rearranging and simplifying equations is often the key to unlocking solutions. By setting the equation equal to zero, we create a scenario where we are essentially finding the roots or x-intercepts of the corresponding quadratic function, which provides a deeper understanding of the solutions we are seeking.
Simplifying the Equation
Before applying the quadratic formula or completing the square, it's often beneficial to simplify the equation if possible. Notice that all the coefficients in our equation, 3x² - 18x - 42 = 0, are divisible by 3. Dividing the entire equation by 3 simplifies it without changing the solutions:
(3x² - 18x - 42) / 3 = 0 / 3
This simplifies to:
x² - 6x - 14 = 0
Now, our equation is simpler, with a = 1, b = -6, and c = -14. Simplifying a quadratic equation before solving it can significantly reduce the complexity of the calculations involved. This step highlights the importance of recognizing common factors and applying the distributive property in reverse to simplify expressions. By dividing all terms by a common factor, we maintain the equation's balance while making the coefficients smaller and easier to work with. This simplification not only reduces the chances of making arithmetic errors but also makes the subsequent steps, such as completing the square or applying the quadratic formula, more manageable. In essence, simplifying the equation is a strategic move that enhances both the efficiency and accuracy of the solution process. Furthermore, this step underscores the general principle in mathematics of seeking the simplest form of an expression or equation to facilitate further analysis and manipulation.
Applying the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our simplified equation, x² - 6x - 14 = 0, we have a = 1, b = -6, and c = -14. Substituting these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)² - 4 * 1 * -14)) / (2 * 1)
Let's break down the calculation step by step. First, we calculate the discriminant, which is the part under the square root:
(-6)² - 4 * 1 * -14 = 36 + 56 = 92
Now, substitute this back into the formula:
x = (6 ± √92) / 2
The quadratic formula is a cornerstone of algebra, providing a universal method for solving quadratic equations regardless of their complexity. Its derivation stems from the process of completing the square, which further highlights its mathematical foundation. The formula's power lies in its ability to handle equations that are difficult or impossible to factor, making it an indispensable tool for mathematicians and scientists alike. The discriminant (b² - 4ac) plays a crucial role in determining the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. This insight into the nature of the roots is invaluable in various applications. Moreover, the quadratic formula is not just a rote memorization task; understanding its derivation and application fosters a deeper appreciation for the interconnectedness of mathematical concepts.
Simplifying the Solution
We have x = (6 ± √92) / 2. To simplify further, we need to simplify the square root. We can factor 92 as 4 * 23. Since √4 = 2, we can rewrite √92 as:
√92 = √(4 * 23) = √4 * √23 = 2√23
Now, substitute this back into our equation:
x = (6 ± 2√23) / 2
We can factor out a 2 from the numerator:
x = 2(3 ± √23) / 2
Finally, we cancel the 2 in the numerator and denominator:
x = 3 ± √23
Simplifying the solution is a critical step in the problem-solving process, ensuring that the answer is presented in its most concise and understandable form. The ability to simplify radicals, such as √92, is a fundamental skill in algebra and requires a solid understanding of factoring and the properties of square roots. By breaking down the radical into its prime factors, we can identify perfect square factors that can be extracted, thereby simplifying the expression. This process not only makes the answer more elegant but also facilitates further calculations or comparisons if needed. Moreover, simplifying the solution often reveals underlying mathematical structures and relationships that might not be immediately apparent in the unsimplified form. The step-by-step simplification process demonstrates the importance of attention to detail and the application of algebraic principles to refine and polish the final answer. In essence, simplifying the solution is the final touch that transforms a raw result into a refined and meaningful answer.
Final Answer
Therefore, the solutions for x in the equation 3x² - 18x + 5 = 47 are:
x = 3 ± √23
This corresponds to option A.
The final answer represents the culmination of the entire problem-solving process, encapsulating all the steps and techniques applied along the way. Arriving at the correct solution not only provides the answer to the specific problem but also reinforces the understanding of the underlying mathematical concepts and methods. The solution x = 3 ± √23 represents two distinct values of x that satisfy the original quadratic equation, highlighting the nature of quadratic equations to have up to two real roots. This final answer is not just a numerical value; it's a testament to the power of algebraic manipulation, the quadratic formula, and the importance of simplification. Moreover, the process of arriving at the final answer underscores the iterative nature of problem-solving, where each step builds upon the previous one, and attention to detail is paramount. In conclusion, the final answer is the ultimate goal of the problem-solving journey, representing the successful application of mathematical knowledge and skills.
Conclusion
Solving the equation 3x² - 18x + 5 = 47 demonstrates the application of key algebraic principles, including simplifying equations, applying the quadratic formula, and simplifying radicals. The solution, x = 3 ± √23, is obtained through a systematic approach, highlighting the importance of each step in the process. This exercise not only provides a solution to a specific problem but also reinforces the broader understanding of quadratic equations and their solutions. Mastering these skills is crucial for success in higher-level mathematics and various scientific and engineering disciplines. The ability to solve quadratic equations is a cornerstone of mathematical literacy, enabling individuals to tackle a wide range of problems in both academic and real-world contexts. Moreover, the problem-solving process itself cultivates critical thinking skills, logical reasoning, and attention to detail, which are valuable assets in any field. By working through this example, we have not only found the solution but also reinforced the fundamental principles that underpin algebraic problem-solving. In conclusion, the journey through this quadratic equation has been a valuable learning experience, highlighting the power and elegance of mathematical techniques.
