Solving 5x² = 35 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. One type of equation that frequently arises is the quadratic equation, which is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This article delves into the process of solving the specific quadratic equation 5x² = 35, providing a step-by-step guide that is accessible to both students and enthusiasts of mathematics. We will explore the underlying principles, the algebraic manipulations involved, and the interpretation of the solutions. Understanding how to solve quadratic equations is crucial for various applications in mathematics, physics, engineering, and other fields. Mastering this skill not only enhances your mathematical prowess but also equips you with a powerful tool for problem-solving in diverse contexts. So, let's embark on this journey of unraveling the solution to 5x² = 35, and along the way, gain a deeper appreciation for the elegance and utility of quadratic equations. This equation might seem simple at first glance, but it encapsulates the core concepts of quadratic equations and their solutions. By understanding the process of solving this equation, you'll gain a solid foundation for tackling more complex quadratic equations in the future. Our approach will be methodical and clear, ensuring that each step is well-explained and easy to follow. We will also highlight the key principles and techniques involved, so you can apply them to other similar problems. Remember, the goal is not just to find the solutions, but to understand the underlying mathematical concepts. This understanding will empower you to solve a wide range of quadratic equations and related problems.

Isolating the x² Term: The First Key Step

To solve the equation 5x² = 35, our initial focus is on isolating the x² term. This means we want to get x² by itself on one side of the equation. The equation 5x² = 35 indicates that 5 is multiplying x². To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 5. This ensures that the equation remains balanced, maintaining the equality. Dividing both sides by 5, we get (5x²)/5 = 35/5. On the left side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with x². On the right side, 35 divided by 5 equals 7. Therefore, the equation simplifies to x² = 7. This step is crucial because it transforms the original equation into a simpler form that is easier to solve. Isolating the squared term is a common strategy in solving quadratic equations, and it often paves the way for the next steps in the solution process. It's important to remember that whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the balance. This principle is fundamental to solving any algebraic equation. Now that we have isolated x², we can proceed to the next step, which involves finding the values of x that satisfy the equation x² = 7. This will lead us to the solutions of the original quadratic equation 5x² = 35. The act of isolating the variable is a cornerstone of algebraic manipulation, and it's a skill that you'll use repeatedly in various mathematical contexts. By mastering this technique, you'll be well-equipped to tackle a wide range of equations and problems.

Finding the Square Root: Unveiling the Solutions

Having isolated x² in the equation x² = 7, the next step is to find the square root of both sides. This is because the square root operation is the inverse of squaring, and it will allow us to isolate x. The square root of x² is x, but it's crucial to remember that taking the square root of a number yields both a positive and a negative solution. This is because both the positive and negative values, when squared, will result in the original number. In this case, the square root of 7 has two solutions: √7 and -√7. Therefore, the solutions to the equation x² = 7 are x = √7 and x = -√7. These are the exact solutions. If we need approximate solutions, we can use a calculator to find the decimal approximations of √7 and -√7. The square root of 7 is approximately 2.646, so the approximate solutions are x ≈ 2.646 and x ≈ -2.646. It's important to express the solutions in both exact and approximate forms, as the context of the problem may dictate which form is more appropriate. The exact solutions provide a precise representation of the values, while the approximate solutions offer a more practical understanding of their magnitude. Understanding the concept of square roots and their relationship to squaring is essential for solving quadratic equations. The fact that there are two solutions, a positive and a negative, is a key characteristic of quadratic equations. This is because the square of both a positive and a negative number is positive. Therefore, when we take the square root to solve for x, we must consider both possibilities. This step highlights the importance of careful attention to detail when working with mathematical operations. Forgetting the negative solution is a common mistake, but by remembering that both positive and negative values can satisfy the equation, we can ensure that we find all the solutions.

Solutions and Verification: Ensuring Accuracy

Now that we have found the solutions x = √7 and x = -√7, it's crucial to verify them. This step ensures that our solutions are accurate and that we haven't made any mistakes in our calculations. To verify the solutions, we substitute each value of x back into the original equation, 5x² = 35, and check if the equation holds true. Let's first substitute x = √7 into the equation: 5(√7)² = 5 * 7 = 35. This confirms that x = √7 is indeed a solution. Next, let's substitute x = -√7 into the equation: 5(-√7)² = 5 * 7 = 35. This also confirms that x = -√7 is a solution. Since both solutions satisfy the original equation, we can confidently conclude that they are correct. Verification is a critical step in problem-solving, not just in mathematics, but in any field. It helps to identify and correct errors, ensuring the accuracy and reliability of the results. In the context of solving equations, verification is particularly important because it's easy to make mistakes in the algebraic manipulations. By substituting the solutions back into the original equation, we can catch any errors and avoid drawing incorrect conclusions. The process of verification also reinforces our understanding of the relationship between the solutions and the original equation. It helps us to see why the solutions work and how they satisfy the equation's conditions. This deeper understanding enhances our problem-solving skills and our overall mathematical literacy. In this case, the verification process has confirmed that the solutions x = √7 and x = -√7 are indeed correct, giving us confidence in our solution to the equation 5x² = 35. This meticulous approach to problem-solving is a hallmark of mathematical excellence, and it's a habit that will serve you well in your studies and beyond. By consistently verifying your solutions, you can develop a greater sense of accuracy and precision in your work.

Conclusion: Mastering Quadratic Equations

In conclusion, we have successfully solved the quadratic equation 5x² = 35 by following a clear and methodical approach. We began by isolating the x² term, then took the square root of both sides, remembering to consider both positive and negative solutions. Finally, we verified our solutions to ensure their accuracy. The solutions to the equation 5x² = 35 are x = √7 and x = -√7, or approximately x ≈ 2.646 and x ≈ -2.646. This process demonstrates the fundamental principles of solving quadratic equations, which are applicable to a wide range of similar problems. Mastering quadratic equations is a crucial skill in mathematics, as they appear in various contexts, from algebra and calculus to physics and engineering. By understanding the steps involved in solving quadratic equations, you gain a powerful tool for problem-solving and a deeper appreciation for the elegance and utility of mathematics. The ability to manipulate equations, isolate variables, and find solutions is a cornerstone of mathematical literacy. It's a skill that will serve you well in your academic pursuits and in your professional life. Quadratic equations are not just abstract mathematical concepts; they have real-world applications in areas such as physics, engineering, economics, and computer science. By mastering the techniques for solving them, you open doors to a wider range of problem-solving opportunities. The journey of solving 5x² = 35 has provided a valuable learning experience. We've not only found the solutions but also gained a deeper understanding of the underlying mathematical concepts. This understanding will empower you to tackle more complex quadratic equations and related problems with confidence and competence. Remember, the key to success in mathematics is practice and perseverance. The more you work with quadratic equations and other mathematical concepts, the more proficient you will become. So, continue to explore, learn, and challenge yourself, and you'll find that the world of mathematics is full of fascinating and rewarding discoveries.