Solving X² - 2X + 1 = 0 Using Bhaskara's Formula A Comprehensive Guide
Introduction: Unveiling the Secrets of Quadratic Equations
In the realm of mathematics, quadratic equations hold a special place. These equations, characterized by the presence of a variable raised to the power of two, pop up in various real-world scenarios, from calculating projectile trajectories to modeling growth and decay. Understanding how to solve quadratic equations is therefore a fundamental skill. One of the most powerful tools in our arsenal for tackling these equations is the Bhaskara's formula, also known as the quadratic formula. This formula provides a systematic way to find the roots, or solutions, of any quadratic equation in the standard form of ax² + bx + c = 0. This article delves into solving the specific quadratic equation X² - 2X + 1 = 0 using Bhaskara's formula, providing a step-by-step guide and ensuring a comprehensive understanding of the process.
The beauty of Bhaskara's formula lies in its universality. Regardless of the complexity of the coefficients 'a', 'b', and 'c', the formula provides a reliable pathway to finding the solutions. This contrasts with other methods, such as factoring, which may not always be readily applicable. Mastering the use of Bhaskara's formula empowers you to confidently solve a wide range of quadratic equations, making it an indispensable tool in your mathematical toolkit. We will not only demonstrate the application of the formula but also provide insights into the nature of the solutions, helping you interpret the results in the context of the original problem. So, let's embark on this journey of unraveling the mysteries of quadratic equations and mastering Bhaskara's formula.
Bhaskara's Formula: Your Key to Quadratic Solutions
At the heart of solving quadratic equations lies Bhaskara's formula. This formula provides a direct method for finding the roots of any quadratic equation in the standard form ax² + bx + c = 0. The formula itself is expressed as:
X = (-b ± √(b² - 4ac)) / 2a
Where:
- X represents the roots of the equation (the values of X that make the equation true).
- a, b, and c are the coefficients of the quadratic equation.
- The symbol '±' indicates that there are potentially two solutions, one obtained by adding the square root term and the other by subtracting it.
The expression inside the square root, b² - 4ac, is known as the discriminant (represented by the Greek letter delta, Δ). The discriminant plays a crucial role in determining the nature of the roots:
- If Δ > 0, the equation has two distinct real roots.
- If Δ = 0, the equation has one real root (a repeated root).
- If Δ < 0, the equation has two complex roots.
Understanding the discriminant is key to interpreting the solutions obtained from Bhaskara's formula. It provides valuable information about the number and type of roots without even fully solving the equation. By first calculating the discriminant, we can anticipate the nature of the solutions and approach the problem with a clearer understanding. Bhaskara's formula is more than just a formula; it's a gateway to understanding the deeper properties of quadratic equations and their solutions. Its power lies in its ability to handle any quadratic equation, regardless of the complexity of its coefficients. This makes it an indispensable tool for mathematicians, scientists, and engineers alike.
Applying Bhaskara's Formula to X² - 2X + 1 = 0: A Step-by-Step Solution
Now, let's apply Bhaskara's formula to solve the specific equation X² - 2X + 1 = 0. This will provide a concrete example of how the formula works in practice.
Step 1: Identify the coefficients
First, we need to identify the coefficients a, b, and c in the equation. Comparing X² - 2X + 1 = 0 to the standard form ax² + bx + c = 0, we can see that:
- a = 1 (the coefficient of X²)
- b = -2 (the coefficient of X)
- c = 1 (the constant term)
Step 2: Calculate the discriminant (Δ)
The discriminant, Δ, is calculated using the formula Δ = b² - 4ac. Substituting the values we identified in step 1, we get:
Δ = (-2)² - 4 * 1 * 1 = 4 - 4 = 0
The discriminant is 0, which indicates that the equation has one real root (a repeated root).
Step 3: Apply Bhaskara's formula
Now, we apply Bhaskara's formula:
X = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a, b, and Δ, we get:
X = (-(-2) ± √0) / (2 * 1) = (2 ± 0) / 2
Step 4: Calculate the roots
Since the square root of 0 is 0, we have:
X = 2 / 2 = 1
Therefore, the equation has one real root, which is X = 1.
This step-by-step process demonstrates the clear and methodical approach offered by Bhaskara's formula. By systematically identifying the coefficients, calculating the discriminant, and applying the formula, we can confidently arrive at the solution. The fact that the discriminant is zero in this case is significant, indicating a repeated root. This means that the quadratic equation represents a perfect square trinomial, which we will discuss further in the next section.
Understanding the Solution Set: S = {1}
In the previous section, we successfully applied Bhaskara's formula to the quadratic equation X² - 2X + 1 = 0 and found that it has one real root, X = 1. This means that the equation is satisfied only when X is equal to 1. Therefore, the solution set for the equation is a set containing only the element 1, which is represented as S = {1}.
The solution set, often denoted by 'S', is a concise way of expressing all the possible values of the variable that satisfy the equation. In this case, since there is only one solution, the solution set is a singleton set. The significance of this solution set lies in its direct connection to the roots of the equation. The roots are the values that make the equation true, and the solution set is simply a collection of these roots. Understanding the solution set provides a complete picture of the solutions to the equation.
Furthermore, the fact that the solution set contains only one element is a consequence of the discriminant being zero. As we discussed earlier, a discriminant of zero indicates a repeated root. This means that the quadratic equation represents a perfect square trinomial, which can be factored as (X - 1)² = 0. This factorization directly reveals that X = 1 is the only solution. The solution set S = {1} therefore encapsulates the complete solution to the quadratic equation X² - 2X + 1 = 0, highlighting the importance of Bhaskara's formula and the concept of the discriminant in understanding the nature and number of solutions.
Alternative Solutions and Verifications
While Bhaskara's formula provides a reliable method for solving quadratic equations, it's always beneficial to explore alternative approaches and verify our solutions. In the case of X² - 2X + 1 = 0, there's a particularly elegant alternative method: factoring.
Factoring:
The expression X² - 2X + 1 is a perfect square trinomial. It can be factored as (X - 1)(X - 1) or (X - 1)². Setting this equal to zero, we have:
(X - 1)² = 0
Taking the square root of both sides, we get:
X - 1 = 0
Adding 1 to both sides, we find:
X = 1
This confirms our solution obtained using Bhaskara's formula. Factoring, when applicable, can often be a quicker and more intuitive method for solving quadratic equations. However, Bhaskara's formula remains a powerful tool, especially for equations that are not easily factorable.
Verification:
To verify our solution, we can substitute X = 1 back into the original equation:
(1)² - 2(1) + 1 = 1 - 2 + 1 = 0
Since the equation holds true, we have verified that X = 1 is indeed the correct solution.
This process of alternative solutions and verification is crucial in mathematics. It not only provides confidence in our answer but also deepens our understanding of the underlying concepts. By exploring different methods and verifying our results, we strengthen our problem-solving skills and develop a more robust mathematical intuition. In this case, the agreement between Bhaskara's formula and factoring, along with the successful verification, reinforces the correctness of our solution and highlights the interconnectedness of different mathematical approaches.
Conclusion: Mastering Quadratic Equations with Bhaskara's Formula
In this comprehensive exploration, we've delved into the world of quadratic equations and mastered the application of Bhaskara's formula. We tackled the specific equation X² - 2X + 1 = 0, demonstrating a step-by-step approach to finding its roots. We identified the coefficients, calculated the discriminant, applied the formula, and arrived at the solution X = 1. We further solidified our understanding by examining the solution set S = {1} and exploring the significance of the discriminant in determining the nature of the roots.
Beyond the mechanics of the formula, we emphasized the importance of understanding the underlying concepts. We explored an alternative solution using factoring and verified our result, highlighting the interconnectedness of different mathematical techniques. This holistic approach is key to developing a deep and lasting understanding of mathematics.
Bhaskara's formula is a powerful tool in any mathematician's arsenal. Its ability to solve any quadratic equation, regardless of complexity, makes it an invaluable asset. However, it's crucial to remember that the formula is not just a black box; it's a reflection of deeper mathematical principles. By understanding these principles, we can use Bhaskara's formula not just as a means to an end, but as a gateway to a richer understanding of quadratic equations and their applications in the world around us.
This journey through solving X² - 2X + 1 = 0 with Bhaskara's formula is a testament to the power of systematic problem-solving and the beauty of mathematical elegance. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical challenges and appreciate the intricate patterns that govern our world.
