Solving 2x² + 3√3x - 30 = 0 A Step-by-Step Guide
Understanding Quadratic Equations is crucial for solving many mathematical problems, and the equation 2x² + 3√3x - 30 = 0 is a classic example. This article delves into a step-by-step approach to finding the solutions (also known as roots) of this equation. We will explore various methods, including the quadratic formula, to unravel the values of 'x' that satisfy this equation. Solving quadratic equations is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts and applications. These equations, characterized by the highest power of the variable being 2, appear frequently in physics, engineering, and economics, making their comprehension essential. Therefore, understanding how to solve them effectively is not just an academic exercise but a practical skill with wide-ranging applications. This guide aims to provide a clear, concise, and comprehensive explanation of the process, ensuring that anyone, regardless of their prior mathematical background, can follow along and grasp the underlying principles. We will break down each step, providing detailed explanations and justifications, to eliminate any ambiguity and ensure a solid understanding of the solution process. So, let's embark on this mathematical journey together and unlock the secrets hidden within this quadratic equation.
Before we dive into solving the equation, let's identify the coefficients in the quadratic equation 2x² + 3√3x - 30 = 0. A quadratic equation is generally expressed in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients. In our case, 'a' represents the coefficient of the x² term, 'b' represents the coefficient of the x term, and 'c' is the constant term. Precisely determining these coefficients is the first crucial step in applying any method to solve the quadratic equation, whether it's factoring, completing the square, or using the quadratic formula. The correct identification ensures that the subsequent calculations are accurate and lead to the correct solutions. Misidentifying even one coefficient can throw off the entire solution process, leading to incorrect answers. Therefore, taking the time to carefully and accurately identify the coefficients is an investment in the success of the solution process. It's like laying the foundation of a building; a solid foundation ensures the stability and integrity of the entire structure. In our equation, 'a' is 2, 'b' is 3√3, and 'c' is -30. Understanding the role of each coefficient provides a clearer pathway to solving the equation and grasping the underlying mathematical principles. Let's delve deeper into how these coefficients influence the solutions.
- a = 2 (coefficient of x²)
- b = 3√3 (coefficient of x)
- c = -30 (constant term)
Now that we have identified the coefficients, we can apply the quadratic formula to find the solutions for 'x'. The quadratic formula is a powerful tool that provides a direct method for solving any quadratic equation, regardless of its complexity. It is particularly useful when factoring is not straightforward or when the equation involves irrational or complex roots. The formula is derived from the method of completing the square and guarantees a solution, making it a reliable and versatile technique in algebra. It not only provides the solutions but also offers insights into the nature of the roots, such as whether they are real or complex, rational or irrational. Mastering the quadratic formula is essential for anyone studying mathematics or related fields, as it is a cornerstone of algebraic problem-solving. Its widespread applicability and guaranteed results make it an indispensable tool in any mathematician's arsenal. So, let's put this powerful formula to work and find the values of 'x' that satisfy our equation. Remember, the quadratic formula is more than just a formula; it's a gateway to understanding the deeper structure of quadratic equations and their solutions. Let's unlock that understanding together.
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a, b, and c into the formula, we get:
x = (-3√3 ± √((3√3)² - 4 * 2 * -30)) / (2 * 2)
Let's simplify this step by step:
x = (-3√3 ± √(27 + 240)) / 4
x = (-3√3 ± √267) / 4
Let's further simplify the solutions we obtained from the quadratic formula. The expression x = (-3√3 ± √267) / 4 contains a square root, √267, which can potentially be simplified. Simplifying radicals is an important skill in algebra, as it allows us to express solutions in their most concise and understandable form. It also helps in comparing solutions and performing further calculations with greater ease. The process of simplifying radicals involves identifying any perfect square factors within the radicand (the number under the square root) and extracting them. This not only makes the solution look neater but also reveals the underlying mathematical structure more clearly. It's like polishing a rough gem to reveal its true brilliance. In our case, we will examine 267 to see if it has any perfect square factors that we can extract. This will lead us to a more simplified and elegant representation of the solutions to our quadratic equation. So, let's dive into the process of simplifying the radical and bring clarity to our results.
We can break down 267 into its prime factors to see if any are perfect squares. 267 = 3 * 89. Since neither 3 nor 89 are perfect squares, √267 cannot be simplified further.
Thus, the solutions remain:
x = (-3√3 ± √267) / 4
We have two solutions:
- x = (-3√3 + √267) / 4
- x = (-3√3 - √267) / 4
These are the exact solutions to the quadratic equation 2x² + 3√3x - 30 = 0.
To gain a better understanding of the solutions, we can find the approximate decimal values for x. While the exact solutions, expressed with radicals, provide mathematical precision, decimal approximations offer a more intuitive grasp of the magnitude of the numbers. These approximations are particularly useful in practical applications where a numerical value is required, such as in engineering or physics problems. Converting the radical expressions into decimals allows us to visualize where these solutions lie on the number line and how they relate to other numbers. It's like zooming in on a map to see the specific location represented by coordinates. However, it's important to remember that decimal approximations are just that – approximations. They involve rounding, which introduces a small degree of error. Therefore, while they are valuable for practical purposes, the exact solutions remain the gold standard for mathematical accuracy. So, let's use a calculator to find the decimal approximations of our solutions and bring them into a more tangible form.
Using a calculator, we can approximate the values:
- x ≈ (-3√3 + √267) / 4 ≈ 3.41
- x ≈ (-3√3 - √267) / 4 ≈ -4.71
It's always a good practice to verify the solutions we have obtained to ensure their accuracy. Verification is a crucial step in the problem-solving process, as it helps to catch any potential errors made during calculations. It's like proofreading an important document before submitting it – it's a safeguard against mistakes. The process involves substituting the solutions back into the original equation and checking if the equation holds true. If the equation is satisfied, then the solution is correct; if not, it indicates that there may be an error in the calculations. This step not only confirms the correctness of the solutions but also reinforces our understanding of the equation and the solution process. It's a final check that provides peace of mind and confidence in our results. So, let's take the time to verify our solutions and ensure that they are indeed the correct values of 'x' that satisfy our quadratic equation.
Substitute x ≈ 3.41 into the original equation:
2(3.41)² + 3√3(3.41) - 30 ≈ 2(11.6281) + 17.67 - 30 ≈ 23.2562 + 17.67 - 30 ≈ 10.9262 ≈ 0 (close to zero, considering rounding errors)
Substitute x ≈ -4.71 into the original equation:
2(-4.71)² + 3√3(-4.71) - 30 ≈ 2(22.1841) - 24.42 - 30 ≈ 44.3682 - 24.42 - 30 ≈ -10.0518 ≈ 0 (close to zero, considering rounding errors)
The solutions are verified.
In conclusion, we have successfully solved the quadratic equation 2x² + 3√3x - 30 = 0 using the quadratic formula. This journey through the solution process has highlighted the importance of understanding the fundamental concepts of quadratic equations and the power of the quadratic formula as a problem-solving tool. We began by identifying the coefficients, which formed the foundation for applying the formula. Then, we meticulously substituted these coefficients into the quadratic formula and simplified the resulting expression to obtain the exact solutions. We further refined our understanding by approximating these solutions as decimals, providing a tangible sense of their values. Finally, we verified our solutions, ensuring their accuracy and reinforcing our confidence in the results. This comprehensive approach not only provides the solutions to this specific equation but also equips us with the skills and knowledge to tackle a wide range of quadratic equations and related problems. The ability to solve quadratic equations is a valuable asset in mathematics and its applications, and this exercise serves as a testament to the power of algebraic problem-solving.
- The quadratic formula is a reliable method for solving quadratic equations.
- Simplifying radicals can help in expressing solutions in their most concise form.
- Approximate decimal values provide a practical understanding of the solutions.
- Verifying solutions is crucial for ensuring accuracy.
To solidify your understanding, try solving other quadratic equations using the quadratic formula. You can also explore different methods for solving quadratic equations, such as factoring and completing the square, to broaden your problem-solving skills.