Solving 3p² - 7p + 1 = 4 A Comprehensive Guide
In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, characterized by the presence of a squared variable, frequently appear in various mathematical contexts and real-world applications. This article provides a comprehensive guide on solving the quadratic equation 3p² - 7p + 1 = 4. We will explore the steps involved in transforming the equation into standard form, and then apply different methods for finding the solutions, also known as roots or zeros, of the equation. Understanding how to solve quadratic equations is crucial for students, engineers, scientists, and anyone dealing with mathematical problems regularly.
Before diving into the solution, it's essential to grasp the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. 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 we aim to find. The coefficient 'a' cannot be zero; otherwise, the equation becomes linear.
In our given equation, 3p² - 7p + 1 = 4, we need to first transform it into the standard form to identify the coefficients a, b, and c. This involves rearranging the equation by subtracting 4 from both sides, which gives us: 3p² - 7p + 1 - 4 = 0, simplifying to 3p² - 7p - 3 = 0. Now, we can clearly see that a = 3, b = -7, and c = -3. Recognizing these coefficients is a key step in applying various solution methods, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach, and understanding when to use each one can greatly streamline the problem-solving process.
To solve the equation 3p² - 7p + 1 = 4, the initial step is to bring it into the standard quadratic form: ax² + bx + c = 0. This form allows us to easily identify the coefficients a, b, and c, which are crucial for applying various solution methods such as factoring, completing the square, or using the quadratic formula. Start by subtracting 4 from both sides of the equation: 3p² - 7p + 1 - 4 = 4 - 4. This simplifies to: 3p² - 7p - 3 = 0. Now, we have the equation in the standard form.
In this standard form, we can identify the coefficients as follows: a = 3 (the coefficient of p²), b = -7 (the coefficient of p), and c = -3 (the constant term). These values are essential for further analysis and solving the equation. Recognizing the coefficients correctly is the foundation for choosing the appropriate method to find the roots. For instance, knowing these values helps in determining whether the equation can be easily factored or if we need to resort to the quadratic formula. Understanding and correctly transforming the equation to standard form is a critical skill in solving any quadratic equation effectively.
There are several methods to solve quadratic equations, each with its own advantages and applications. The three primary methods are factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific equation and the ease with which one can apply each technique. Factoring is generally the quickest method when the quadratic expression can be easily factored into two binomials. Completing the square is a more systematic approach that can be used for any quadratic equation, and it is particularly useful when the equation is not easily factored. The quadratic formula is a universally applicable method that provides a direct solution for any quadratic equation, regardless of its factorability.
Understanding these different methods allows us to tackle a wide range of quadratic equations efficiently. The method of factoring involves rewriting the quadratic expression as a product of two binomials. This method is efficient when the roots are rational numbers. Completing the square involves transforming the equation into a perfect square trinomial, which can then be easily solved. This method is beneficial for understanding the structure of quadratic equations and deriving the quadratic formula. The quadratic formula provides a straightforward solution by substituting the coefficients a, b, and c into the formula, making it a reliable tool for solving any quadratic equation. Each method enhances our problem-solving toolkit, providing flexibility in addressing diverse mathematical challenges.
Factoring is one of the most straightforward methods for solving quadratic equations, but it's not always applicable. This method involves expressing the quadratic equation as a product of two binomials. For example, if we have an equation in the form of ax² + bx + c = 0, we aim to find two binomials (px + q)(rx + s) such that when multiplied, they give us the original quadratic equation. The values of p, q, r, and s must be chosen carefully to ensure that the product matches the original equation.
However, factoring is only feasible when the quadratic expression can be easily decomposed into two binomials with integer coefficients. This often depends on whether the discriminant (b² - 4ac) is a perfect square. If the discriminant is not a perfect square, factoring may not be the most efficient method. In our equation, 3p² - 7p - 3 = 0, attempting to factor directly might be challenging because there are no obvious integer factors that fit the required conditions. When factoring is not immediately apparent, other methods like completing the square or using the quadratic formula become more practical alternatives. Therefore, while factoring is a valuable tool, it's essential to recognize its limitations and be prepared to use other techniques when necessary. Understanding when to apply factoring and when to opt for other methods is a key skill in solving quadratic equations efficiently.
Completing the square is a powerful technique for solving quadratic equations, especially when factoring is not straightforward. This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. The process consists of several key steps, ensuring the equation is manipulated correctly to achieve the desired form. First, divide the entire equation by the coefficient of the p² term if it's not 1. In our case, the equation is 3p² - 7p - 3 = 0, so we divide by 3 to get: p² - (7/3)p - 1 = 0.
Next, move the constant term to the right side of the equation: p² - (7/3)p = 1. To complete the square, we need to add and subtract the square of half the coefficient of the p term. The coefficient of p is -7/3, so half of it is -7/6, and its square is (7/6)² = 49/36. Add 49/36 to both sides of the equation: p² - (7/3)p + 49/36 = 1 + 49/36. The left side is now a perfect square trinomial, which can be written as (p - 7/6)². The right side simplifies to 36/36 + 49/36 = 85/36. So, the equation becomes: (p - 7/6)² = 85/36. Taking the square root of both sides gives us: p - 7/6 = ±√(85/36), which simplifies to p - 7/6 = ±√85 / 6. Finally, solve for p: p = 7/6 ± √85 / 6. This method provides a systematic approach to solving quadratic equations, particularly useful when factoring is not easily applicable. Completing the square enhances our understanding of quadratic equations and their structure, offering a reliable technique for finding solutions.
The quadratic formula is a universally applicable method for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct way to find the roots of the equation, regardless of whether it can be factored or not. The quadratic formula is given by: x = [-b ± √(b² - 4ac)] / (2a). This formula is derived from the method of completing the square and is a cornerstone in algebra.
To apply the quadratic formula to our equation, 3p² - 7p - 3 = 0, we identify the coefficients as a = 3, b = -7, and c = -3. Substituting these values into the formula, we get: p = [-(-7) ± √((-7)² - 4(3)(-3))] / (2(3)). Simplifying further, we have: p = [7 ± √(49 + 36)] / 6, which becomes p = [7 ± √85] / 6. Thus, the two solutions for p are p = (7 + √85) / 6 and p = (7 - √85) / 6. The quadratic formula ensures that we can solve any quadratic equation, making it an essential tool in mathematics. Its ability to provide solutions regardless of the equation's complexity makes it a go-to method for many. Understanding and applying the quadratic formula is a fundamental skill for anyone working with algebraic equations.
Now, let's apply the methods discussed to our specific equation, 3p² - 7p - 3 = 0. We've already transformed the equation into standard form, identifying a = 3, b = -7, and c = -3. First, we'll consider factoring. As mentioned earlier, factoring is not straightforward in this case because there are no obvious integer factors that fit. Therefore, we move on to the other methods.
Next, we apply the method of completing the square. Dividing by 3, we get p² - (7/3)p - 1 = 0. Moving the constant term, we have p² - (7/3)p = 1. Completing the square involves adding and subtracting (7/6)² = 49/36. Thus, we get (p - 7/6)² = 1 + 49/36 = 85/36. Taking the square root, p - 7/6 = ±√85 / 6, leading to p = 7/6 ± √85 / 6. Finally, we use the quadratic formula: p = [-(-7) ± √((-7)² - 4(3)(-3))] / (2(3)) = [7 ± √85] / 6, giving the same solutions as completing the square. Thus, the solutions are p = (7 + √85) / 6 and p = (7 - √85) / 6. This demonstrates the consistency of different methods in solving quadratic equations, reinforcing the importance of choosing the most efficient approach for each problem.
After applying the quadratic formula and completing the square, we have determined the solutions to the equation 3p² - 7p - 3 = 0. The solutions are given by p = (7 + √85) / 6 and p = (7 - √85) / 6. These values represent the points where the quadratic function intersects the x-axis, also known as the roots or zeros of the equation.
The two solutions indicate that there are two distinct real roots for this quadratic equation. These solutions can be approximated to decimal values for practical applications. The numerical approximations help in visualizing the positions of these roots on a graph and in understanding their significance in real-world contexts. Understanding the nature and values of these solutions is crucial in various fields, including engineering, physics, and economics, where quadratic equations model numerous phenomena. The solutions provide critical insights into the behavior of the system or process being modeled, making the ability to find and interpret these solutions an essential skill.
In conclusion, solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications. This article has detailed the process of solving the quadratic equation 3p² - 7p + 1 = 4, from transforming it into standard form to applying various solution methods. We explored the methods of factoring, completing the square, and the quadratic formula, ultimately demonstrating the use of the quadratic formula to find the solutions p = (7 + √85) / 6 and p = (7 - √85) / 6.
Understanding these methods not only equips you with the tools to solve quadratic equations but also enhances your problem-solving abilities in various mathematical contexts. The ability to choose the most efficient method for a given equation is a valuable skill, and this article has provided a comprehensive guide to making that choice. Whether you are a student, engineer, or anyone working with mathematical models, mastering the techniques for solving quadratic equations is essential for success. The knowledge and skills gained here will serve as a strong foundation for tackling more complex mathematical challenges in the future.
