Solve 4x² + 3 = 199 Equation: Step-by-Step Solution

In this article, we will delve into a detailed explanation of how to solve the equation 4x² + 3 = 199. This is a classic algebraic problem that involves isolating the variable x and finding its possible values. Whether you're a student brushing up on your algebra skills or simply curious about equation-solving techniques, this guide will provide a step-by-step approach to understanding and solving this type of problem. We'll break down each step, ensuring clarity and comprehension. So, let's dive in and tackle this equation!

1. Understanding the Equation and Initial Steps

At the heart of our task is the equation 4x² + 3 = 199. This equation is a quadratic equation, recognizable by the presence of the x² term. Quadratic equations often have two solutions, which represent the values of x that make the equation true. Our primary goal is to isolate x² on one side of the equation, paving the way to find the values of x. To begin, we need to isolate the term containing x², which is 4x². This involves using the fundamental principles of algebra, where we perform the same operation on both sides of the equation to maintain balance. The first step is to eliminate the constant term (+3) on the left side. We achieve this by subtracting 3 from both sides of the equation. This operation is crucial because it simplifies the equation and brings us closer to isolating x². After performing this subtraction, the equation transforms, setting the stage for the next steps in our solution process. This initial step is a testament to the importance of understanding and applying algebraic principles in solving equations. By carefully isolating terms, we create a pathway toward finding the solutions for x.

2. Isolating the x² Term

Continuing from our initial step, we now have the equation 4x² = 196. Our next objective is to isolate the x² term completely. Currently, x² is being multiplied by 4. To isolate x², we need to undo this multiplication. The inverse operation of multiplication is division, so we divide both sides of the equation by 4. This is a critical step because it further simplifies the equation and brings us closer to determining the value(s) of x. When we divide both sides by 4, we are left with x² on the left side and a numerical value on the right side. This value represents the square of x. By isolating x², we've essentially set up the equation in a form where we can directly solve for x by taking the square root. This step highlights the significance of using inverse operations to solve algebraic equations. Each operation we perform is carefully chosen to isolate the variable and ultimately find its value(s). The process of isolating x² is a key concept in solving quadratic equations and demonstrates the power of algebraic manipulation.

3. Solving for x by Taking the Square Root

After isolating x², we arrive at the equation x² = 49. This equation tells us that x, when multiplied by itself, equals 49. To find the value(s) of x, we need to perform the inverse operation of squaring, which is taking the square root. When we take the square root of a number, we are looking for the value that, when multiplied by itself, gives us that number. However, it's crucial to remember that both positive and negative values can satisfy this condition. For instance, both 7 and -7, when squared, equal 49. Therefore, when solving for x, we must consider both the positive and negative square roots. This is a common point of oversight, but it is essential for finding all possible solutions to the equation. By taking the square root of both sides of the equation, we find that x can be either 7 or -7. These are the two values that, when substituted back into the original equation, will make the equation true. This step underscores the importance of recognizing the dual nature of square roots and their role in solving quadratic equations.

4. Verifying the Solutions

Having found our potential solutions, x = 7 and x = -7, it's crucial to verify their correctness. This step is a fundamental practice in mathematics, ensuring that the solutions we've obtained are indeed valid. To verify, we substitute each value of x back into the original equation, 4x² + 3 = 199, and check if the equation holds true. Let's start with x = 7. Substituting this value into the equation, we get 4(7)² + 3. Evaluating this expression, we find that it equals 199, which matches the right side of the original equation. This confirms that x = 7 is a valid solution. Next, we repeat the process for x = -7. Substituting this value into the equation, we get 4(-7)² + 3. Again, evaluating this expression, we find that it also equals 199. This confirms that x = -7 is also a valid solution. The verification step not only ensures the accuracy of our solutions but also reinforces our understanding of the equation and the solution process. It's a vital step in problem-solving that should never be overlooked.

5. Expressing the Solutions

Now that we have verified both solutions, x = 7 and x = -7, the final step is to express them in the requested format. The instructions specify that the answers should be listed separated by a comma, without using a ± sign, and without including "x=". This format ensures clarity and consistency in the presentation of the solutions. Following these instructions, we can simply write the two values, separated by a comma. This concise presentation allows for easy readability and direct understanding of the solutions. Expressing solutions in the correct format is an important aspect of mathematical communication, ensuring that the answers are clear and readily interpretable. In this case, the format requirement also tests attention to detail, which is a valuable skill in mathematical problem-solving.

6. Conclusion: Summarizing the Solution Process

In conclusion, we've successfully solved the equation 4x² + 3 = 199 by following a systematic, step-by-step approach. We began by understanding the equation and identifying it as a quadratic equation. The first step involved isolating the term containing x² by subtracting 3 from both sides. We then further isolated x² by dividing both sides by 4. This led us to the equation x² = 49. To solve for x, we took the square root of both sides, remembering to consider both positive and negative roots. This gave us two potential solutions: x = 7 and x = -7. We then verified these solutions by substituting them back into the original equation, confirming their validity. Finally, we expressed the solutions in the requested format, listing them separated by a comma. This process demonstrates the power of algebraic manipulation and the importance of understanding inverse operations. By breaking down the problem into smaller, manageable steps, we were able to confidently solve the equation and arrive at the correct solutions. This comprehensive guide provides a clear understanding of how to tackle similar algebraic problems in the future. The solutions are 7, -7.