Solving N^2 + 4 = 40 A Step-by-Step Guide
#Introduction
In the realm of mathematical equations, solving for unknowns is a fundamental skill. This article delves into the step-by-step solution of the equation n^2 + 4 = 40, providing a clear and concise explanation suitable for students and math enthusiasts alike. We will explore the algebraic principles involved, ensuring a thorough understanding of the process. This equation, while seemingly simple, exemplifies the core concepts of algebra and serves as a stepping stone to more complex mathematical problems. The journey of solving equations is not just about finding the answer; it's about understanding the underlying logic and developing problem-solving skills that are applicable in various fields.
The equation n^2 + 4 = 40 is a quadratic equation in disguise. A quadratic equation is generally expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. In our case, we can rearrange the given equation to fit this form. The variable we are solving for is n, and our goal is to isolate n on one side of the equation. Understanding the structure of the equation is the first crucial step in finding the solution. We need to recognize that the n^2 term is the key element, and we will manipulate the equation to isolate this term before solving for n. This involves applying algebraic operations to both sides of the equation to maintain balance and ensure the equality remains valid. The process of solving for n will involve inverse operations, such as subtraction and square root, to undo the operations applied to n. This methodical approach is essential for solving any algebraic equation, regardless of its complexity.
Isolating the n^2 term:
The first step in solving the equation n^2 + 4 = 40 is to isolate the n^2 term. To achieve this, we need to eliminate the constant term, which is 4, from the left side of the equation. We can do this by subtracting 4 from both sides of the equation. This maintains the equality and moves us closer to isolating n^2. The principle behind this operation is that whatever we do to one side of the equation, we must do to the other to keep the equation balanced. Subtracting 4 from both sides gives us:
n^2 + 4 - 4 = 40 - 4
Simplifying this, we get:
n^2 = 36
Now we have successfully isolated the n^2 term on the left side of the equation. This is a significant step forward, as we are now one step closer to solving for n. The next step will involve undoing the square operation to find the value of n. This will require us to take the square root of both sides of the equation, which we will discuss in the following section.
Taking the square root:
Now that we have n^2 = 36, the next step is to find the value of n. Since n is squared, we need to perform the inverse operation, which is taking the square root. Remember that when we take the square root of a number, we need to consider both the positive and negative roots. This is because both the positive and negative values, when squared, will result in the same positive number. In this case, the square root of 36 is both 6 and -6, since 6 * 6 = 36 and (-6) * (-6) = 36. Therefore, we have two possible solutions for n:
n = √36 and n = -√36
This gives us:
n = 6 and n = -6
So, the solutions to the equation n^2 + 4 = 40 are n = 6 and n = -6. It's important to understand that quadratic equations often have two solutions, and this example illustrates that principle. We have successfully found both values of n that satisfy the original equation.
To ensure our solutions are correct, it's always a good practice to verify them by substituting them back into the original equation. This step helps us catch any potential errors made during the solving process. Let's substitute n = 6 and n = -6 into the equation n^2 + 4 = 40.
Verifying n = 6:
Substituting n = 6 into the equation, we get:
(6)^2 + 4 = 40
Simplifying, we have:
36 + 4 = 40
40 = 40
This confirms that n = 6 is indeed a solution to the equation, as it satisfies the equality.
Verifying n = -6:
Now, let's substitute n = -6 into the equation:
(-6)^2 + 4 = 40
Simplifying, we get:
36 + 4 = 40
40 = 40
This also confirms that n = -6 is a solution to the equation. Both solutions, n = 6 and n = -6, satisfy the original equation, which validates our solution process. This step of verification is crucial in mathematics to ensure accuracy and build confidence in the solutions obtained. It reinforces the understanding of the equation and the solution process.
In this article, we have successfully solved the equation n^2 + 4 = 40. We began by understanding the equation and identifying the goal of isolating the variable n. We then followed a step-by-step approach, first isolating the n^2 term by subtracting 4 from both sides of the equation. Next, we took the square root of both sides, remembering to consider both positive and negative roots. This led us to two solutions: n = 6 and n = -6. Finally, we verified our solutions by substituting them back into the original equation, confirming their validity. This process demonstrates the importance of following a systematic approach when solving algebraic equations. Understanding the underlying principles and applying them methodically is key to success in mathematics. The ability to solve such equations is a fundamental skill that forms the basis for more advanced mathematical concepts. This exercise not only provides the solution to a specific problem but also reinforces the general principles of algebraic problem-solving.
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