Solving X² - 2X + 1 = 0 A Comprehensive Guide Using Bhaskara's Formula
Introduction to Quadratic Equations and Bhaskara's Formula
In the realm of algebra, quadratic equations hold a significant position due to their wide applicability in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving a quadratic equation involves finding the values of the variable (usually denoted as 'x') that satisfy the equation. These values are known as the roots or solutions of the equation.
One of the most powerful and widely used methods for solving quadratic equations is Bhaskara's formula. This formula provides a direct way to calculate the roots of any quadratic equation, regardless of the nature of the coefficients. The formula is derived from the method of completing the square and is named after the 12th-century Indian mathematician Bhāskara II, who made significant contributions to algebra and other areas of mathematics.
Understanding Bhaskara's formula is essential for anyone studying algebra or related fields. It not only provides a reliable method for solving quadratic equations but also deepens our understanding of the underlying mathematical principles. In this article, we will delve into the application of Bhaskara's formula by solving a specific quadratic equation: X² - 2X + 1 = 0. We will break down the steps involved, explain the significance of each component of the formula, and discuss the nature of the roots obtained. By the end of this discussion, you will have a clear understanding of how to use Bhaskara's formula to solve quadratic equations and interpret the results effectively.
Understanding Bhaskara's Formula: A Detailed Explanation
Bhaskara's formula is a cornerstone in solving quadratic equations, providing a systematic approach to finding the roots. To fully appreciate its power, it's crucial to understand the formula itself and its components. The formula is expressed as follows:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- 'x' represents the roots of the quadratic equation.
- 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0.
- The symbol '±' indicates that there are typically two roots, 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, often denoted by the Greek letter delta (Δ). The discriminant plays a critical role in determining the nature of the roots:
- If Δ > 0: The equation has two distinct real roots.
- If Δ = 0: The equation has exactly one real root (a repeated root).
- If Δ < 0: The equation has two complex roots.
Let's break down the formula step by step to understand its application. The first part, -b / 2a, gives the x-coordinate of the vertex of the parabola represented by the quadratic equation. This point is the axis of symmetry for the parabola and provides valuable information about the roots.
The second part, √(b² - 4ac) / 2a, determines the distance from the vertex to the roots. The square root term, √(b² - 4ac), is the crucial element that dictates the nature of the roots. As mentioned earlier, the sign of the discriminant (b² - 4ac) determines whether the roots are real and distinct, real and repeated, or complex.
In essence, Bhaskara's formula is a compact and comprehensive solution for quadratic equations. It encapsulates the relationship between the coefficients of the equation and its roots, providing a direct pathway to finding the solutions. Understanding each component of the formula and its significance is essential for its effective application.
Solving X² - 2X + 1 = 0 Using Bhaskara's Formula: Step-by-Step
Now, let's apply Bhaskara's formula to solve the quadratic equation X² - 2X + 1 = 0. This will provide a practical demonstration of how the formula works and how to interpret the results.
Step 1: Identify the Coefficients
The first step is to identify the coefficients 'a', 'b', and 'c' from the given equation. In the equation X² - 2X + 1 = 0:
- a = 1 (the coefficient of X²)
- b = -2 (the coefficient of X)
- c = 1 (the constant term)
Step 2: Calculate the Discriminant (Δ)
The next step is to calculate the discriminant (Δ) using the formula Δ = b² - 4ac. Substituting the values of a, b, and c, we get:
Δ = (-2)² - 4 * 1 * 1 Δ = 4 - 4 Δ = 0
The discriminant is 0, which indicates that the equation has exactly one real root (a repeated root).
Step 3: Apply Bhaskara's Formula
Now, we apply Bhaskara's formula to find the root(s):
x = (-b ± √(b² - 4ac)) / 2a
Substitute the values of a, b, and Δ:
x = (-(-2) ± √0) / (2 * 1)
x = (2 ± 0) / 2
Step 4: Simplify and Find the Root(s)
Since the square root of 0 is 0, we have:
x = 2 / 2
x = 1
Therefore, the quadratic equation X² - 2X + 1 = 0 has one real root, which is x = 1. This means the equation has a repeated root at x = 1.
Step 5: Verification (Optional)
To verify the solution, we can substitute x = 1 back into the original equation:
(1)² - 2(1) + 1 = 0 1 - 2 + 1 = 0 0 = 0
The equation holds true, confirming that x = 1 is indeed the root of the equation.
By following these steps, we have successfully solved the quadratic equation X² - 2X + 1 = 0 using Bhaskara's formula. The discriminant helped us determine the nature of the roots, and the formula provided a direct method for finding the solution.
Interpreting the Results: Understanding the Root of X² - 2X + 1 = 0
In the previous section, we used Bhaskara's formula to solve the quadratic equation X² - 2X + 1 = 0 and found that it has one real root, x = 1. Now, let's delve deeper into interpreting this result and understanding its implications.
The fact that the equation has only one real root (a repeated root) is significant. It indicates that the graph of the quadratic equation, which is a parabola, touches the x-axis at only one point. In other words, the vertex of the parabola lies on the x-axis at x = 1.
To further illustrate this, let's consider the factored form of the quadratic equation. X² - 2X + 1 can be factored as (X - 1)(X - 1) or (X - 1)². This form clearly shows that x = 1 is a repeated root because the factor (X - 1) appears twice.
Graphically, the parabola represented by the equation Y = X² - 2X + 1 opens upwards (since the coefficient of X² is positive) and has its vertex at the point (1, 0). The vertex is the minimum point of the parabola, and since it lies on the x-axis, it represents the only solution to the equation Y = 0.
The repeated root also has implications in terms of the quadratic formula. When the discriminant (Δ) is equal to 0, as in this case, the term √(b² - 4ac) becomes 0. This means that the ± part of Bhaskara's formula vanishes, and we are left with only one solution, x = -b / 2a. This is precisely what happened in our calculation, where we obtained x = 1 as the only root.
In summary, the solution x = 1 for the equation X² - 2X + 1 = 0 represents a repeated real root. This corresponds to a parabola that touches the x-axis at a single point, indicating that the vertex of the parabola lies on the x-axis at x = 1. Understanding the nature of the roots and their graphical representation provides a comprehensive understanding of the quadratic equation and its solutions.
Alternative Methods for Solving X² - 2X + 1 = 0: Factoring and Completing the Square
While Bhaskara's formula provides a reliable method for solving quadratic equations, it's not the only approach available. In certain cases, alternative methods like factoring and completing the square can be more efficient or provide additional insights into the equation's structure. Let's explore these methods in the context of the equation X² - 2X + 1 = 0.
1. Factoring:
Factoring involves expressing the quadratic equation as a product of two linear factors. This method is particularly effective when the quadratic expression can be easily factored. In the case of X² - 2X + 1 = 0, we can recognize that the expression is a perfect square trinomial. It can be factored as:
X² - 2X + 1 = (X - 1)(X - 1) = (X - 1)²
Setting this equal to zero, we have:
(X - 1)² = 0
Taking the square root of both sides:
X - 1 = 0
Solving for X:
X = 1
This method directly yields the solution x = 1, which is a repeated root, as we found using Bhaskara's formula. Factoring provides a clear and concise way to solve this particular equation, highlighting its perfect square nature.
2. Completing the Square:
Completing the square is another method for solving quadratic equations that involves transforming the equation into a form where one side is a perfect square. This method is particularly useful for deriving Bhaskara's formula itself. To complete the square for X² - 2X + 1 = 0, we follow these steps:
- Ensure the coefficient of X² is 1 (which it already is in this case).
- Take half of the coefficient of X (-2), which is -1, and square it: (-1)² = 1.
- Add and subtract this value within the equation: X² - 2X + 1 + 0 = 0
Notice that X² - 2X + 1 is already a perfect square trinomial, so we can rewrite the equation as:
(X - 1)² = 0
Taking the square root of both sides:
X - 1 = 0
Solving for X:
X = 1
Again, we arrive at the solution x = 1, confirming our previous results. Completing the square provides a systematic way to transform the equation and solve for the roots, even when factoring is not immediately obvious.
In conclusion, while Bhaskara's formula is a powerful tool for solving quadratic equations, factoring and completing the square offer alternative approaches that can be more efficient in certain cases. These methods also provide valuable insights into the structure and properties of quadratic equations.
Significance of Bhaskara's Formula in Mathematics and Beyond
Bhaskara's formula is not just a method for solving quadratic equations; it's a fundamental concept with far-reaching implications in mathematics and various other fields. Its significance stems from its generality, its connection to other mathematical concepts, and its practical applications in diverse areas.
One of the primary reasons for the importance of Bhaskara's formula is its generality. It can be used to solve any quadratic equation, regardless of the nature of the coefficients. Whether the coefficients are integers, fractions, decimals, or even irrational numbers, Bhaskara's formula will always provide the roots, provided they exist in the real or complex number system. This universality makes it an indispensable tool in algebra and related fields.
Moreover, Bhaskara's formula is closely related to other fundamental mathematical concepts. As we discussed earlier, it is derived from the method of completing the square, which is a powerful technique for solving quadratic equations and other types of algebraic problems. The formula also connects to the concept of the discriminant, which provides valuable information about the nature of the roots without actually solving the equation. The discriminant helps us determine whether the roots are real and distinct, real and repeated, or complex.
Furthermore, Bhaskara's formula has significant applications beyond pure mathematics. Quadratic equations arise in numerous real-world scenarios, including physics, engineering, economics, and computer science. For example, in physics, quadratic equations are used to model projectile motion, the trajectory of objects under the influence of gravity. In engineering, they are used in circuit analysis, structural design, and control systems. In economics, quadratic equations can model cost functions, revenue functions, and profit maximization problems.
The formula's ability to solve these equations makes it an essential tool for scientists, engineers, economists, and other professionals. Whether it's calculating the trajectory of a rocket, designing a bridge, or optimizing a business strategy, Bhaskara's formula provides a reliable and efficient way to find solutions.
In addition, the study of Bhaskara's formula and quadratic equations lays the foundation for understanding more advanced mathematical concepts. It introduces students to the idea of solving equations, working with algebraic expressions, and interpreting mathematical results. These skills are crucial for success in higher-level mathematics courses, such as calculus, linear algebra, and differential equations.
Conclusion: Mastering Quadratic Equations with Bhaskara's Formula
In this comprehensive discussion, we have explored the power and versatility of Bhaskara's formula in solving quadratic equations. We began by introducing the formula and its components, emphasizing the significance of the discriminant in determining the nature of the roots. We then applied the formula step-by-step to solve the specific equation X² - 2X + 1 = 0, demonstrating how to identify the coefficients, calculate the discriminant, and find the roots.
We further interpreted the results, understanding that the single real root x = 1 represents a repeated root, which corresponds to the vertex of the parabola touching the x-axis at that point. We also explored alternative methods for solving the equation, such as factoring and completing the square, highlighting their efficiency and providing additional insights into the equation's structure.
Finally, we discussed the broader significance of Bhaskara's formula in mathematics and beyond. Its generality, its connection to other mathematical concepts, and its practical applications in various fields make it an indispensable tool for students, scientists, engineers, and professionals alike.
Mastering Bhaskara's formula is a crucial step in developing a strong foundation in algebra and mathematics. It not only provides a reliable method for solving quadratic equations but also enhances our understanding of the underlying mathematical principles. By understanding the formula, its components, and its applications, we can confidently tackle a wide range of problems and appreciate the beauty and power of mathematics.
As you continue your mathematical journey, remember that Bhaskara's formula is a valuable tool in your arsenal. Practice applying it to various quadratic equations, explore its connections to other mathematical concepts, and appreciate its significance in solving real-world problems. With dedication and practice, you can master quadratic equations and unlock new levels of mathematical understanding.
