Solving 3x(x + 6) = -10 The Most Direct Method
In this article, we will tackle the equation 3x(x + 6) = -10 using the most direct method available. This involves expanding the equation, rearranging it into a standard quadratic form, and then applying the quadratic formula to find the solutions. Our goal is to present the solution in its exact, most simplified form, utilizing the ± symbol if there are two solutions. Understanding the nuances of solving quadratic equations is fundamental in mathematics, and this step-by-step guide will help clarify the process.
Understanding Quadratic Equations
Before diving into the specifics of our equation, it’s crucial to understand quadratic equations in general. 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 ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros, and they represent the values of x that satisfy the equation. These solutions can be real or complex numbers. Solving quadratic equations is a fundamental skill in algebra, with applications spanning various fields such as physics, engineering, and economics. The solutions to these equations often represent critical points, such as the trajectory of a projectile or the equilibrium points in a system. Understanding the nature and behavior of quadratic equations is essential for problem-solving in many mathematical and real-world contexts. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and is suitable for different types of quadratic equations. The quadratic formula, in particular, is a universal method that can be applied to any quadratic equation, regardless of its coefficients. Recognizing the structure and properties of quadratic equations is the first step towards mastering algebraic problem-solving.
Steps to Solve Quadratic Equations
Solving quadratic equations involves a systematic approach to find the values of the variable that satisfy the equation. First, it is essential to rearrange the equation into the standard form ax² + bx + c = 0. This involves expanding any products, combining like terms, and moving all terms to one side of the equation, leaving zero on the other side. Once the equation is in standard form, you can identify the coefficients a, b, and c. Next, you can choose a method to solve the equation. Factoring is a quick method if the quadratic expression can be easily factored. Completing the square is another method that involves transforming the quadratic expression into a perfect square trinomial. The quadratic formula is a universal method that can be used to solve any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a). Applying the quadratic formula involves substituting the values of a, b, and c into the formula and simplifying the expression. The discriminant, which is the expression b² - 4ac under the square root, determines the nature of the roots. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If it is negative, there are two complex roots. After obtaining the solutions, it is crucial to simplify them as much as possible and express them in their exact form, avoiding decimal approximations unless required. This systematic approach ensures accuracy and efficiency in solving quadratic equations.
Solving the Equation 3x(x + 6) = -10
Now, let's apply these principles to our specific equation: 3x(x + 6) = -10. The first step is to expand the left side of the equation by distributing the 3x term. This gives us 3x² + 18x = -10. To bring the equation into the standard quadratic form, we need to move the -10 to the left side, which yields 3x² + 18x + 10 = 0. Now we have a quadratic equation in the form ax² + bx + c = 0, where a = 3, b = 18, and c = 10. With the equation in standard form, we can proceed to solve it using the quadratic formula. The quadratic formula is a reliable method for finding the solutions of any quadratic equation. Applying the quadratic formula in a structured manner helps in avoiding errors and ensures that we obtain the correct solutions. The steps involved in using the quadratic formula include substituting the coefficients into the formula, simplifying the expression, and then interpreting the results based on the discriminant. This method is particularly useful when the quadratic equation is not easily factorable, making it a versatile tool in solving a wide range of problems. Understanding the quadratic formula and its application is essential for mastering quadratic equations and their solutions.
Applying the Quadratic Formula
To solve the quadratic equation 3x² + 18x + 10 = 0, we will use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). We have already identified that a = 3, b = 18, and c = 10. Substituting these values into the formula, we get:
x = (-18 ± √(18² - 4 * 3 * 10)) / (2 * 3)
Now we simplify the expression step by step. First, calculate the value inside the square root:
18² = 324
4 * 3 * 10 = 120
So, the expression under the square root becomes:
324 - 120 = 204
Now we substitute this back into the formula:
x = (-18 ± √204) / 6
Next, we simplify the square root. We look for perfect square factors of 204. The prime factorization of 204 is 2² * 3 * 17, so we can rewrite √204 as √(2² * 51) = 2√51. Substituting this back into the equation, we get:
x = (-18 ± 2√51) / 6
Finally, we simplify the entire expression by dividing each term by 2:
x = (-9 ± √51) / 3
Thus, the solutions to the equation are x = (-9 + √51) / 3 and x = (-9 - √51) / 3. These are the exact, most simplified forms of the solutions.
Expressing the Solution with the ± Symbol
When we have two solutions that differ only by a sign, it is often convenient to express them using the ± symbol. In our case, the solutions are x = (-9 + √51) / 3 and x = (-9 - √51) / 3. We can combine these into a single expression using the ± symbol:
x = (-9 ± √51) / 3
This notation succinctly represents both solutions, making it a compact and efficient way to present the final answer. The use of the ± symbol is a standard practice in mathematics, especially when dealing with quadratic equations and other situations where multiple solutions exist. It not only saves space but also clearly indicates the symmetrical nature of the solutions. Understanding and using this notation is essential for effective communication in mathematical contexts. It allows for a more concise and understandable representation of solutions, particularly in more complex problems where the solutions might be lengthy or involve multiple terms. The ability to express solutions in this manner is a valuable skill in algebra and beyond.
Importance of Exact Form
In mathematics, expressing solutions in their exact form is crucial for several reasons. Exact solutions preserve the precision and integrity of the answer, avoiding the rounding errors that can occur when using decimal approximations. For example, leaving the solution in terms of square roots or fractions maintains the true value, which can be essential in further calculations or applications. Decimal approximations, while sometimes useful for practical purposes, can introduce inaccuracies that propagate through subsequent steps, leading to significant errors in the final result. Moreover, exact forms often reveal underlying mathematical structures and relationships that might be obscured by decimal approximations. For instance, a solution involving a square root might indicate a geometric connection or a specific property of the problem. In many scientific and engineering applications, precision is paramount, and using exact forms ensures the highest level of accuracy. Additionally, expressing solutions in exact form allows for easier comparison and simplification, especially when dealing with multiple solutions or complex expressions. The preference for exact forms underscores the importance of mathematical rigor and the pursuit of truth in its most precise form. Therefore, maintaining exact solutions is a fundamental aspect of mathematical problem-solving and analysis.
Conclusion
In conclusion, we have successfully solved the equation 3x(x + 6) = -10 using a direct method that involved expanding the equation, rearranging it into standard quadratic form, and applying the quadratic formula. We found the solutions to be x = (-9 ± √51) / 3, expressing them in their exact, most simplified form using the ± symbol. This step-by-step approach highlights the importance of understanding quadratic equations and the techniques used to solve them. The ability to solve quadratic equations is a fundamental skill in mathematics with broad applications in various fields. By mastering these techniques, one can tackle a wide range of problems involving quadratic relationships. The emphasis on exact solutions underscores the importance of precision and mathematical rigor in problem-solving. This process not only provides the answer but also enhances the understanding of the underlying mathematical principles, paving the way for more advanced mathematical studies and applications. The skills and insights gained from solving quadratic equations are invaluable assets in any mathematical endeavor.