Solving 3x² - 4x = 15 A Step-by-Step Guide
Introduction: Mastering Quadratic Equations
In the realm of mathematics, quadratic equations hold a position of fundamental importance. These equations, characterized by their second-degree polynomial form, appear in various scientific and engineering applications. Understanding how to solve them is a crucial skill for anyone delving into advanced mathematical concepts. This article will delve into the process of solving the specific quadratic equation 3x² - 4x = 15. We'll explore the various methods available, including factoring, completing the square, and the quadratic formula, ultimately demonstrating a step-by-step solution using the quadratic formula.
Quadratic equations are expressed in the general form of ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. The solutions, also known as roots or zeros, represent the values of 'x' that satisfy the equation. Finding these solutions involves manipulating the equation to isolate 'x'. One common approach involves rearranging the equation into the standard form and then employing a suitable solution method. Factoring, a technique that breaks down the quadratic expression into simpler linear factors, is effective when the equation can be easily factored. Completing the square, another method, involves transforming the equation into a perfect square trinomial, allowing for a more direct solution. However, the quadratic formula, a universal solution derived from completing the square, provides a reliable method for solving any quadratic equation, regardless of its factorability. This formula offers a direct way to find the roots by plugging in the coefficients 'a', 'b', and 'c'. For the given equation, 3x² - 4x = 15, we will first rearrange it into the standard form and then apply the quadratic formula to find the solutions for 'x'. Understanding the nuances of each method empowers us to tackle a wide range of quadratic equations effectively. We will dissect the process, highlighting key steps and providing clear explanations to ensure a comprehensive understanding. By mastering these techniques, you'll gain a valuable tool for solving mathematical problems and applying them to real-world scenarios.
Understanding the Equation: 3x² - 4x = 15
To effectively solve the equation 3x² - 4x = 15, the initial step involves transforming it into the standard quadratic form, which is ax² + bx + c = 0. This form is crucial because it allows us to readily identify the coefficients 'a', 'b', and 'c', which are essential for applying methods like the quadratic formula. In the given equation, we observe that the constant term is on the right side. Therefore, we need to subtract 15 from both sides to bring the equation into the standard form. This process ensures that all terms are on one side, leaving zero on the other, a prerequisite for most solution techniques.
Subtracting 15 from both sides of the equation yields 3x² - 4x - 15 = 0. Now, the equation is in the standard quadratic form, and we can easily identify the coefficients. Here, 'a' is the coefficient of x², which is 3. 'b' is the coefficient of x, which is -4. And 'c' is the constant term, which is -15. Identifying these coefficients is a fundamental step as they are the building blocks for applying the quadratic formula and other solution methods. Understanding the role of each coefficient is vital for correctly substituting them into the formula and obtaining accurate solutions. The coefficient 'a' determines the parabola's direction (whether it opens upwards or downwards) and its width. The coefficient 'b' influences the position of the parabola's axis of symmetry, while 'c' represents the y-intercept of the parabola. With the equation now in standard form and the coefficients identified, we are well-prepared to proceed with solving for 'x'. The subsequent steps will involve choosing an appropriate solution method, such as factoring, completing the square, or the quadratic formula, based on the equation's characteristics and the solver's preference. This foundational understanding sets the stage for a systematic and accurate solution process, ensuring that we arrive at the correct roots of the equation. The next section will delve into the application of the quadratic formula to this specific equation.
Method 1: Applying the Quadratic Formula
The quadratic formula stands as a cornerstone in solving quadratic equations. It provides a universal method for finding the roots of any equation in the form ax² + bx + c = 0. The formula itself is expressed as: x = (-b ± √(b² - 4ac)) / (2a). This formula is derived from the method of completing the square and offers a direct approach to finding the solutions without the need for factoring or other manipulations. To apply the quadratic formula, we first need to identify the coefficients 'a', 'b', and 'c' from our equation, which we have already done in the previous section.
In our case, the equation is 3x² - 4x - 15 = 0, so a = 3, b = -4, and c = -15. Now, we substitute these values into the quadratic formula: x = (-(-4) ± √((-4)² - 4 * 3 * -15)) / (2 * 3). The next step involves simplifying the expression. First, we deal with the negative signs and the terms inside the square root. The expression becomes: x = (4 ± √(16 + 180)) / 6. We then simplify the expression under the square root: x = (4 ± √196) / 6. Since the square root of 196 is 14, we further simplify the expression to: x = (4 ± 14) / 6. Now, we have two possible solutions, one with the plus sign and one with the minus sign. The first solution is x = (4 + 14) / 6 = 18 / 6 = 3. The second solution is x = (4 - 14) / 6 = -10 / 6 = -5/3. Therefore, the solutions to the equation 3x² - 4x = 15 are x = 3 and x = -5/3. These values represent the points where the parabola represented by the equation intersects the x-axis. The quadratic formula's power lies in its ability to handle any quadratic equation, regardless of whether it can be factored easily or not. It's a reliable tool in mathematical problem-solving and a fundamental concept in algebra. In summary, by correctly identifying the coefficients and substituting them into the formula, we can efficiently find the roots of any quadratic equation.
Method 2: Exploring Factoring (Alternative Approach)
While the quadratic formula provides a direct solution, factoring offers an alternative approach to solving quadratic equations. Factoring involves expressing the quadratic expression as a product of two linear factors. This method is particularly efficient when the quadratic expression can be easily factored. However, not all quadratic equations are easily factorable, making the quadratic formula a more universally applicable method. To attempt factoring, we need to rewrite the equation 3x² - 4x - 15 = 0 in the form (px + q)(rx + s) = 0, where p, q, r, and s are constants.
Finding these constants involves identifying two numbers that multiply to give the product of the leading coefficient (3) and the constant term (-15), which is -45, and add up to the middle coefficient (-4). These two numbers are -9 and 5 because (-9) * 5 = -45 and (-9) + 5 = -4. Using these numbers, we can rewrite the middle term (-4x) as -9x + 5x. The equation then becomes 3x² - 9x + 5x - 15 = 0. Now, we can factor by grouping. We group the first two terms and the last two terms: (3x² - 9x) + (5x - 15) = 0. From the first group, we can factor out 3x, and from the second group, we can factor out 5. This gives us: 3x(x - 3) + 5(x - 3) = 0. Notice that (x - 3) is a common factor. We can factor it out, resulting in: (3x + 5)(x - 3) = 0. Now, we have the equation in factored form. To find the solutions, we set each factor equal to zero: 3x + 5 = 0 and x - 3 = 0. Solving the first equation, 3x + 5 = 0, we subtract 5 from both sides to get 3x = -5, and then divide by 3 to get x = -5/3. Solving the second equation, x - 3 = 0, we add 3 to both sides to get x = 3. Therefore, the solutions obtained by factoring are x = -5/3 and x = 3, which match the solutions obtained using the quadratic formula. Factoring, when applicable, can be a quicker method than the quadratic formula. It provides a deeper understanding of the structure of the quadratic expression and its relationship to its roots. However, it requires a bit of intuition and pattern recognition, which may not always be straightforward. In cases where factoring is not immediately apparent, the quadratic formula serves as a reliable alternative.
Conclusion: Mastering Quadratic Equation Solutions
In conclusion, we have successfully solved the quadratic equation 3x² - 4x = 15 using two distinct methods: the quadratic formula and factoring. Both methods led us to the same solutions, x = 3 and x = -5/3, demonstrating the consistency and reliability of these techniques. The quadratic formula, a universal tool, provides a direct solution for any quadratic equation, regardless of its complexity or factorability. It involves substituting the coefficients 'a', 'b', and 'c' into the formula and simplifying the resulting expression. Factoring, on the other hand, is an efficient method when the quadratic expression can be easily factored into linear factors. It requires identifying the correct factors and setting each factor equal to zero to find the solutions.
Understanding both methods empowers you to choose the most appropriate approach based on the specific equation at hand. For equations that are easily factorable, factoring can be a quicker and more intuitive method. However, for equations that are not easily factorable or when a guaranteed solution is desired, the quadratic formula is the preferred choice. Mastering quadratic equations is a fundamental skill in mathematics, with applications spanning various fields, including physics, engineering, and computer science. The ability to solve these equations opens doors to understanding and solving complex problems in these disciplines. Whether you choose to use the quadratic formula, factoring, or another method, the key is to practice and develop a strong understanding of the underlying principles. With practice, you'll become proficient in solving quadratic equations and gain a valuable tool for your mathematical toolkit. This comprehensive exploration of the solution process, from understanding the equation to applying different methods, equips you with the knowledge and skills to tackle quadratic equations confidently and effectively. Remember, each method offers a unique perspective on solving these equations, and mastering both enhances your problem-solving abilities in mathematics.
