Solving X² - 18x + 81 = 0 Finding The Values Of X

In the realm of mathematics, solving equations is a fundamental skill. Quadratic equations, in particular, hold a significant place due to their wide applicability in various fields. This article delves into the process of finding the value(s) of x that satisfy the quadratic equation x² - 18x + 81 = 0. We will explore different methods to solve this equation, including factoring, using the quadratic formula, and completing the square. By understanding these techniques, you'll be equipped to tackle a wide range of quadratic equations.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of x that make the equation true. In our case, the equation x² - 18x + 81 = 0 fits this form, with a = 1, b = -18, and c = 81. Before we dive into the solution, it is very crucial to understand the properties of the equation that will help you identify the correct method to use to arrive at the final solution. You will encounter different quadratic equations that can be solved using factoring, completing the square or the quadratic formula. Choosing the correct method will save you time and unnecessary steps.

Method 1: Factoring

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be easily expressed as a product of two binomials. The key idea behind factoring is to reverse the process of expansion. We look for two numbers that multiply to give the constant term (c) and add up to the coefficient of the linear term (b). For the equation x² - 18x + 81 = 0, we need to find two numbers that multiply to 81 and add up to -18. After some consideration, we can identify these numbers as -9 and -9. Therefore, we can rewrite the equation as (x - 9)(x - 9) = 0. This means (x - 9)² = 0. To find the value of x, we set each factor equal to zero: x - 9 = 0. Solving for x, we get x = 9. In this case, we have a repeated root, meaning the equation has only one distinct solution. Factoring method is the easiest and most preferred method when the quadratic equation has integer roots. It is fast and less prone to errors when done carefully. You need to carefully analyze the equation to identify if the equation can be factored easily. If it is not straight forward, then the other methods can be explored.

Method 2: Using the Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations. It provides a direct solution for x regardless of whether the equation can be factored or not. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / 2a. In our equation, x² - 18x + 81 = 0, we have a = 1, b = -18, and c = 81. Plugging these values into the quadratic formula, we get: x = (18 ± √((-18)² - 4 * 1 * 81)) / (2 * 1). Simplifying the expression under the square root, we have: x = (18 ± √(324 - 324)) / 2. This simplifies to: x = (18 ± √0) / 2. Further simplification gives us: x = 18 / 2 = 9. Again, we obtain the solution x = 9. The quadratic formula is a powerful tool, especially when dealing with quadratic equations that are difficult to factor. It guarantees a solution, even if the roots are complex numbers. However, it involves more calculations compared to factoring, so it's generally preferred when factoring is not straightforward. The discriminant, b² - 4ac, within the quadratic formula, provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, the equation has one real root (a repeated root). If it's negative, the equation has two complex roots.

Method 3: Completing the Square

Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial on one side. To complete the square for the equation x² - 18x + 81 = 0, we first ensure that the coefficient of x² is 1, which it already is in this case. Next, we take half of the coefficient of the x term (-18), which is -9, and square it, which gives us 81. Notice that 81 is already the constant term in our equation. This means our equation is already a perfect square trinomial. We can rewrite the equation as (x - 9)² = 0. Taking the square root of both sides, we get: x - 9 = 0. Solving for x, we find x = 9. Completing the square is a valuable technique, particularly when the quadratic equation is not easily factorable. It's also a fundamental concept in understanding the derivation of the quadratic formula. However, for equations where the coefficient of x² is not 1 or when the constant term is not a perfect square, completing the square can involve more intricate steps. Understanding the geometric interpretation of completing the square can further enhance your understanding of this method. It involves visualizing the quadratic expression as an area and manipulating it to form a square.

The Solution

Using all three methods – factoring, the quadratic formula, and completing the square – we arrive at the same solution: x = 9. This indicates that the quadratic equation x² - 18x + 81 = 0 has one real solution, which is a repeated root. This solution corresponds to the point where the parabola represented by the equation touches the x-axis. The graph of the equation y = x² - 18x + 81 is a parabola that opens upwards. Since the equation has only one real root, the vertex of the parabola lies on the x-axis at the point (9, 0). This is a characteristic feature of quadratic equations with repeated roots. The vertex represents the minimum value of the parabola, which in this case is 0.

Conclusion

In this article, we explored the process of finding the values of x that satisfy the quadratic equation x² - 18x + 81 = 0. We demonstrated three different methods – factoring, using the quadratic formula, and completing the square – all of which led to the same solution: x = 9. Understanding these methods provides you with a comprehensive toolkit for solving quadratic equations. Factoring is often the quickest method when applicable, while the quadratic formula provides a universal solution. Completing the square offers valuable insights into the structure of quadratic equations. By mastering these techniques, you'll be well-equipped to tackle a wide range of mathematical problems involving quadratic equations. Remember to always check your solutions by substituting them back into the original equation to ensure accuracy. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and computer science. Continued practice and exploration will solidify your understanding and enhance your problem-solving abilities.