Introduction: Navigating Quadratic Equations
In the realm of mathematics, quadratic equations hold a significant position, often appearing in various real-world applications and theoretical problems. Understanding how to solve these equations is crucial for students and professionals alike. This article delves into the process of finding the values of x that satisfy the equation x² - 6x + 9 = 25. We will explore different methods, providing a step-by-step guide to ensure clarity and comprehension. By mastering these techniques, you'll gain a solid foundation for tackling more complex mathematical challenges.
Understanding the Quadratic Equation
Before we embark on solving the equation, let's first understand the basics of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 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. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and disadvantages, and the choice of method often depends on the specific equation at hand.
Factoring Method
Factoring is a method of solving quadratic equations by expressing the quadratic expression as a product of two linear factors. This method is particularly effective when the quadratic expression can be easily factored. To factor a quadratic expression, we need to find two numbers that multiply to give the constant term and add up to give the coefficient of the linear term. In our case, we will explore how factoring can be applied to solve the given equation, highlighting its efficiency and potential limitations.
Completing the Square Method
Completing the square is another powerful technique for solving quadratic equations. This method involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root of both sides. Completing the square is particularly useful when the quadratic expression cannot be easily factored. This method provides a systematic approach to solving quadratic equations and is a fundamental concept in algebra. Understanding completing the square not only helps in solving equations but also provides insights into the structure of quadratic expressions.
Quadratic Formula Method
The quadratic formula is a universal method for solving quadratic equations. It provides a direct solution for x regardless of whether the quadratic expression can be factored or not. The quadratic formula is derived from the method of completing the square and is given by: x = (-b ± √(b² - 4ac)) / 2a. This formula is a cornerstone of algebra and is widely used in various mathematical and scientific applications. By using the quadratic formula, we can efficiently find the roots of any quadratic equation, making it an indispensable tool in problem-solving.
Solving the Equation x² - 6x + 9 = 25
Now, let's apply these methods to solve the equation x² - 6x + 9 = 25. We will demonstrate how each method can be used to arrive at the solutions for x. This hands-on approach will solidify your understanding of the techniques and their practical application.
Method 1: Factoring
The given equation is x² - 6x + 9 = 25. Before we can factor, we need to rewrite the equation in the standard form ax² + bx + c = 0. Subtracting 25 from both sides, we get:
x² - 6x + 9 - 25 = 0 x² - 6x - 16 = 0
Now, we look for two numbers that multiply to -16 and add up to -6. These numbers are -8 and 2. So, we can factor the quadratic expression as:
(x - 8)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 8 = 0 or x + 2 = 0 x = 8 or x = -2
Therefore, the solutions are x = 8 and x = -2.
Method 2: Completing the Square
To solve x² - 6x + 9 = 25 by completing the square, we first observe that the left-hand side is already a perfect square trinomial:
(x - 3)² = 25
Taking the square root of both sides, we get:
x - 3 = ±√25 x - 3 = ±5
Now, we solve for x:
x = 3 + 5 or x = 3 - 5 x = 8 or x = -2
Again, we find the solutions to be x = 8 and x = -2.
Method 3: Quadratic Formula
To use the quadratic formula, we need to rewrite the equation in the standard form x² - 6x - 16 = 0. Here, a = 1, b = -6, and c = -16. Plugging these values into the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a x = (-(-6) ± √((-6)² - 4(1)(-16))) / 2(1) x = (6 ± √(36 + 64)) / 2 x = (6 ± √100) / 2 x = (6 ± 10) / 2
Now, we find the two solutions:
x = (6 + 10) / 2 = 16 / 2 = 8 x = (6 - 10) / 2 = -4 / 2 = -2
Thus, the solutions are x = 8 and x = -2, confirming our previous results.
Verifying the Solutions
To ensure our solutions are correct, we can substitute them back into the original equation x² - 6x + 9 = 25. This step is crucial in confirming the validity of our solutions and preventing errors.
Verifying x = 8
Substituting x = 8 into the equation:
(8)² - 6(8) + 9 = 25 64 - 48 + 9 = 25 25 = 25
The equation holds true, so x = 8 is a valid solution.
Verifying x = -2
Substituting x = -2 into the equation:
(-2)² - 6(-2) + 9 = 25 4 + 12 + 9 = 25 25 = 25
The equation also holds true for x = -2, confirming it as a valid solution.
Conclusion: Mastering Quadratic Equation Solutions
In conclusion, the values of x that satisfy the equation x² - 6x + 9 = 25 are x = 8 and x = -2. We have demonstrated three different methods to solve this quadratic equation: factoring, completing the square, and using the quadratic formula. Each method provides a unique approach to finding the solutions, and understanding all three methods equips you with a comprehensive toolkit for tackling quadratic equations. Remember to always verify your solutions by substituting them back into the original equation to ensure accuracy. By mastering these techniques, you'll be well-prepared to solve a wide range of mathematical problems involving quadratic equations.
This article has provided a detailed explanation of how to solve the quadratic equation x² - 6x + 9 = 25. By understanding the different methods and their applications, you can confidently approach similar problems and enhance your mathematical skills. Keep practicing and exploring different types of quadratic equations to further solidify your knowledge.