Solving 2x² - 2√2x + 1 = 0 With The Quadratic Formula A Step-by-Step Guide
In this article, we will delve into the process of solving the quadratic equation 2x² - 2√2x + 1 = 0 using the quadratic formula. Understanding how to apply the quadratic formula is a fundamental skill in algebra, allowing us to find the roots (or solutions) of any quadratic equation in the form ax² + bx + c = 0. This particular equation involves a radical term, making it a good example to illustrate the power and versatility of the quadratic formula. We will break down each step, ensuring a clear and comprehensive understanding of the solution.
The quadratic formula is a mathematical expression that provides the solutions to any quadratic equation. It is given by:
x = [-b ± √(b² - 4ac)] / 2a
Where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. Before we jump into the solution, let’s understand why the quadratic formula is so important and how it’s derived. Quadratic equations appear in various fields, including physics, engineering, and economics, making their solutions critical in these disciplines. The quadratic formula itself is derived by completing the square on the general form of a quadratic equation, which is a technique that transforms the equation into a perfect square trinomial, making it easier to solve. The discriminant, b² - 4ac, plays a crucial role in determining 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); and if it is negative, there are two complex roots.
In the subsequent sections, we will first identify the coefficients in our given equation, then substitute these values into the quadratic formula, and finally simplify the expression to arrive at our solutions. By the end of this article, you should be confident in your ability to apply the quadratic formula to solve any quadratic equation, regardless of its complexity. Whether you are a student looking to improve your algebra skills or someone seeking a refresher on mathematical concepts, this guide will provide a step-by-step approach to mastering this essential formula. So, let’s dive in and unravel the solutions to this quadratic equation together.
Identifying Coefficients
The first step in solving the quadratic equation 2x² - 2√2x + 1 = 0 using the quadratic formula is to correctly identify the coefficients a, b, and c. These coefficients correspond to the general form of a quadratic equation, which is ax² + bx + c = 0. Precisely identifying these values is crucial because any error in this step will propagate through the rest of the solution, leading to an incorrect answer.
In our given equation, 2x² - 2√2x + 1 = 0, we can directly map the coefficients as follows:
- The coefficient a is the number multiplying the x² term. In this case, a = 2.
- The coefficient b is the number multiplying the x term. Here, b = -2√2.
- The coefficient c is the constant term, which is the term without any x. Thus, c = 1.
It's important to pay close attention to the signs of the coefficients. In our case, b is negative, which is a common area for mistakes. Misidentifying a negative sign can significantly alter the outcome. Once we have correctly identified these coefficients, we can confidently substitute them into the quadratic formula.
The quadratic formula, as we stated earlier, is:
x = [-b ± √(b² - 4ac)] / 2a
Now that we know a = 2, b = -2√2, and c = 1, we are ready for the next step, which involves substituting these values into the formula. This substitution is a mechanical process, but precision is key. Each value must be placed correctly in the formula to ensure the accuracy of our calculations. This careful identification and substitution lay the groundwork for solving the equation and finding its roots.
Before moving on, it’s worth reiterating the importance of this initial step. The coefficients are the foundation upon which the entire solution is built. A clear understanding of how to identify a, b, and c will not only help in solving this particular equation but will also prove invaluable for tackling any quadratic equation you encounter. The next step will involve plugging these values into the quadratic formula and simplifying, which will bring us closer to our final solution. So, let’s ensure we have a solid grasp of this foundation before proceeding further.
Substituting Coefficients into the Quadratic Formula
With the coefficients a = 2, b = -2√2, and c = 1 correctly identified, the next crucial step is to substitute these values into the quadratic formula. This process involves carefully replacing the variables a, b, and c in the formula with their respective numerical values. The quadratic formula, as a reminder, is given by:
x = [-b ± √(b² - 4ac)] / 2a
Let's perform the substitution step-by-step to ensure clarity and accuracy. We will replace each variable in the formula with its corresponding value:
- Replace b with -2√2: -(-2√2) becomes 2√2
- Replace b² with (-2√2)²: (-2√2)² simplifies to 8
- Replace 4ac with 4 * 2 * 1: 4 * 2 * 1 equals 8
- Replace 2a with 2 * 2: 2 * 2 equals 4
After performing these substitutions, our equation now looks like this:
x = [2√2 ± √(8 - 8)] / 4
This step is critical because it sets the stage for simplifying the expression and eventually finding the solutions for x. It’s a mechanical process, but precision is paramount. Double-checking each substitution can prevent errors that would otherwise lead to an incorrect final answer. The substitution process effectively translates the abstract coefficients into concrete numbers within the framework of the quadratic formula.
Notice how the negative sign in front of b becomes positive when we substitute -2√2. This is a common area where mistakes can occur, so it’s worth highlighting the importance of paying close attention to these details. The term inside the square root, b² - 4ac, is known as the discriminant, and it provides valuable information about the nature of the roots of the quadratic equation. In our case, the discriminant is 8 - 8 = 0, which indicates that we will have exactly one real root (or a repeated root).
Now that we have successfully substituted the coefficients into the quadratic formula, the next step is to simplify the expression. This will involve performing the arithmetic operations, simplifying the square root, and reducing the fraction to its simplest form. By carefully simplifying the equation, we will arrive at the solution for x, thus completing the process of solving the quadratic equation. So, let's proceed to the simplification stage with confidence, knowing that we have laid a solid foundation with our accurate substitutions.
Simplifying the Expression
Having successfully substituted the coefficients into the quadratic formula, the next critical step is simplifying the expression. This involves performing the arithmetic operations within the formula to arrive at a more manageable form, eventually leading us to the solution for x. Our equation, after substitution, looks like this:
x = [2√2 ± √(8 - 8)] / 4
The first part of the simplification process involves dealing with the square root term. Notice that inside the square root, we have 8 - 8, which equals 0. Therefore, the square root term simplifies to √0, which is simply 0.
x = [2√2 ± 0] / 4
This greatly simplifies our expression. Now we have:
x = [2√2] / 4
Since adding or subtracting 0 doesn't change the value, we can drop the ± 0 part. Next, we need to simplify the fraction. Both the numerator and the denominator have a common factor of 2. We can divide both the numerator and the denominator by 2:
x = (2√2) / 4 = √2 / 2
So, after simplifying the expression, we find that x = √2 / 2. This is the simplified form of the solution to our quadratic equation. The simplification process is a critical step because it transforms a complex-looking expression into a clear and concise solution. Each arithmetic operation must be performed accurately, and any common factors should be canceled out to reach the simplest form of the answer.
In this particular case, the simplification was straightforward because the discriminant (the term inside the square root) was zero. This meant that we had only one real root, which made the final simplification step relatively simple. However, in other quadratic equations, the simplification process might involve dealing with square roots of non-zero numbers, complex numbers, or more complex fractions. The principles remain the same: perform the operations in the correct order, simplify the square roots, and reduce the fractions.
Now that we have simplified the expression and arrived at our solution, x = √2 / 2, we can confidently state the solution to the quadratic equation 2x² - 2√2x + 1 = 0. This solution represents the value of x that satisfies the equation, meaning that if we substitute this value back into the original equation, it will hold true. The next section will summarize our solution and discuss its implications in the context of the original equation. We will also reflect on the steps we took to arrive at the solution, reinforcing the importance of each stage in the process.
Solution and Conclusion
After carefully applying the quadratic formula and simplifying the resulting expression, we have found the solution to the quadratic equation 2x² - 2√2x + 1 = 0. The solution we obtained is:
x = √2 / 2
This result means that the value x = √2 / 2 is the root of the given quadratic equation. In other words, if we substitute √2 / 2 back into the original equation, the equation will be satisfied. This can be verified by plugging x = √2 / 2 into the original equation:
2(√2 / 2)² - 2√2(√2 / 2) + 1 = 0 2(2 / 4) - 2(2 / 2) + 1 = 0 1 - 2 + 1 = 0 0 = 0
As we can see, the equation holds true, confirming that x = √2 / 2 is indeed the correct solution.
In summary, solving a quadratic equation using the quadratic formula involves several key steps. First, we identified the coefficients a, b, and c from the equation. Then, we substituted these values into the quadratic formula. After substitution, we simplified the expression by performing the necessary arithmetic operations, including simplifying the square root and reducing the fraction. In this case, the discriminant (b² - 4ac) was zero, which indicated that there is exactly one real root (a repeated root). This simplified the process, allowing us to arrive at the final solution efficiently.
The quadratic formula is a powerful tool for solving quadratic equations, and it is especially useful when factoring or completing the square is not straightforward. Understanding how to correctly apply the formula, including identifying coefficients, substituting values, and simplifying expressions, is a fundamental skill in algebra. This example demonstrates the step-by-step process of using the quadratic formula and highlights the importance of accuracy and attention to detail.
In conclusion, the solution to the quadratic equation 2x² - 2√2x + 1 = 0 is x = √2 / 2. This exercise underscores the effectiveness of the quadratic formula in solving such equations and reinforces the importance of a systematic approach to mathematical problem-solving. By mastering these techniques, you can confidently tackle a wide range of quadratic equations and other algebraic problems.
