Solving X² - 6x + 9 = 25 A Comprehensive Guide To Quadratic Equations

In this article, we will delve into the process of solving a quadratic equation. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields, including mathematics, physics, engineering, and economics. A typical quadratic equation is expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we aim to solve for. Our specific problem involves finding the values of x that satisfy the equation x² - 6x + 9 = 25. To solve this equation effectively, we will explore different techniques, including factoring, completing the square, and using the quadratic formula. Each method offers a unique approach to finding the roots or solutions of the equation. Understanding these methods not only helps in solving this particular problem but also provides a strong foundation for tackling more complex quadratic equations in the future. By the end of this discussion, you will have a clear understanding of how to manipulate and solve quadratic equations, enabling you to apply these skills in various mathematical and real-world contexts. So, let's embark on this mathematical journey and unravel the solutions to our equation, x² - 6x + 9 = 25, by carefully examining each step and method involved. Remember, mastering quadratic equations is a significant step towards advancing your mathematical abilities and problem-solving skills.

Factoring is a powerful technique for solving quadratic equations, especially when the equation can be expressed as a product of two binomials. In this section, we will apply the factoring method to the equation x² - 6x + 9 = 25. The first step in factoring is to rearrange the equation so that it equals zero. This involves subtracting 25 from both sides of the equation, which gives us x² - 6x + 9 - 25 = 0. Simplifying this, we get x² - 6x - 16 = 0. Now, we need to find two numbers that multiply to -16 (the constant term) and add up to -6 (the coefficient of the x term). These numbers are -8 and 2, since (-8) * 2 = -16 and (-8) + 2 = -6. Therefore, we can rewrite the quadratic equation as (x - 8)(x + 2) = 0. To find the values of x that satisfy this equation, we set each factor equal to zero. This gives us two equations: x - 8 = 0 and x + 2 = 0. Solving the first equation, x - 8 = 0, we add 8 to both sides, which gives us x = 8. Solving the second equation, x + 2 = 0, we subtract 2 from both sides, which gives us x = -2. Thus, the solutions to the equation x² - 6x + 9 = 25 are x = 8 and x = -2. Factoring is a useful method because it simplifies the equation into a more manageable form, allowing us to quickly identify the roots. However, it's important to note that factoring is not always straightforward and may not be applicable to all quadratic equations. In such cases, other methods like completing the square or using the quadratic formula may be more appropriate.

The quadratic formula is a versatile method for solving any quadratic equation, regardless of whether it can be factored or not. This formula is particularly useful when the factoring method proves to be challenging or impossible. The general form of a quadratic equation is ax² + bx + c = 0, and the quadratic formula to find the values of x is given by: x = [-b ± √(b² - 4ac)] / (2a). To apply the quadratic formula to our equation, x² - 6x + 9 = 25, we first need to rewrite it in the standard form by subtracting 25 from both sides, resulting in x² - 6x - 16 = 0. Now, we can identify the coefficients: a = 1 (the coefficient of x²), b = -6 (the coefficient of x), and c = -16 (the constant term). Substituting these values into the quadratic formula, we get: x = [-(-6) ± √((-6)² - 4 * 1 * (-16))] / (2 * 1). Simplifying this expression, we have: x = [6 ± √(36 + 64)] / 2. Further simplification gives us: x = [6 ± √100] / 2. Since the square root of 100 is 10, the equation becomes: x = [6 ± 10] / 2. This leads to two possible solutions for x. First, we take the positive root: x = (6 + 10) / 2 = 16 / 2 = 8. Second, we take the negative root: x = (6 - 10) / 2 = -4 / 2 = -2. Therefore, the solutions to the equation x² - 6x + 9 = 25, as found using the quadratic formula, are x = 8 and x = -2. The quadratic formula provides a reliable and systematic approach to solving quadratic equations, making it an essential tool in algebra. Its ability to handle any quadratic equation, regardless of its factorability, makes it a valuable method for solving complex mathematical problems.

Completing the square is another powerful technique for solving quadratic equations, offering a systematic way to rewrite the equation into a perfect square trinomial. This method is particularly useful for equations that do not factor easily. To apply completing the square to the equation x² - 6x + 9 = 25, we first need to rewrite the equation in the form (x - h)² = k. The given equation is x² - 6x + 9 = 25. Notice that the left side of the equation, x² - 6x + 9, is already a perfect square trinomial. It can be factored as (x - 3)². So, the equation can be written as (x - 3)² = 25. Now, to solve for x, we take the square root of both sides of the equation. This gives us √(x - 3)² = ±√25, which simplifies to x - 3 = ±5. We now have two separate equations to solve: x - 3 = 5 and x - 3 = -5. Solving the first equation, x - 3 = 5, we add 3 to both sides, which gives us x = 5 + 3 = 8. Solving the second equation, x - 3 = -5, we add 3 to both sides, which gives us x = -5 + 3 = -2. Thus, the solutions to the equation x² - 6x + 9 = 25, as found by completing the square, are x = 8 and x = -2. Completing the square is a valuable technique not only for solving quadratic equations but also for understanding the structure of quadratic expressions and their graphical representations. It provides a clear pathway to the solutions and enhances one's understanding of algebraic manipulations.

After exploring three different methods for solving the quadratic equation x² - 6x + 9 = 25—factoring, using the quadratic formula, and completing the square—it is essential to compare these approaches and verify the solutions we obtained. Each method offers a unique way to tackle the equation, and understanding their strengths and weaknesses can help us choose the most efficient method for different scenarios. Factoring, as we saw, involves rewriting the quadratic equation as a product of two binomials. This method is often the quickest when the equation can be easily factored, but it may not be suitable for all quadratic equations. The quadratic formula, on the other hand, is a more general approach that can be applied to any quadratic equation, regardless of its factorability. It provides a systematic way to find the solutions, but it can be more computationally intensive than factoring, especially when dealing with complex coefficients. Completing the square is another powerful technique that involves rewriting the equation in a form that allows us to easily take the square root. This method is particularly useful for understanding the structure of quadratic expressions and their graphical representations. In our case, all three methods led us to the same solutions: x = 8 and x = -2. To verify these solutions, we can substitute them back into the original equation, x² - 6x + 9 = 25, to ensure they satisfy the equation. For x = 8, we have: (8)² - 6(8) + 9 = 64 - 48 + 9 = 25, which confirms that x = 8 is a valid solution. For x = -2, we have: (-2)² - 6(-2) + 9 = 4 + 12 + 9 = 25, which also confirms that x = -2 is a valid solution. By verifying our solutions, we ensure the accuracy of our calculations and gain confidence in our problem-solving abilities. This step is crucial in mathematics and helps prevent errors, especially in more complex problems. In conclusion, the solutions to the equation x² - 6x + 9 = 25 are x = 8 and x = -2, and we have verified these solutions using both factoring, the quadratic formula, and completing the square.

In summary, we have successfully determined the values of x in the quadratic equation x² - 6x + 9 = 25 using three distinct methods: factoring, the quadratic formula, and completing the square. Each method provided a unique approach to solving the equation, and all three led us to the same solutions. Through factoring, we rewrote the equation as (x - 8)(x + 2) = 0, which directly gave us the solutions x = 8 and x = -2. By applying the quadratic formula, we systematically calculated the roots using the coefficients of the equation, again arriving at x = 8 and x = -2. Completing the square allowed us to transform the equation into the form (x - 3)² = 25, which, upon taking the square root, also yielded x = 8 and x = -2. The consistency of these results across different methods reinforces the accuracy of our solutions. We further verified these solutions by substituting them back into the original equation, confirming that both x = 8 and x = -2 satisfy the equation x² - 6x + 9 = 25. This comprehensive exploration not only solves the specific problem but also enhances our understanding of quadratic equations and the various techniques available for solving them. The ability to approach a problem from multiple angles and verify the results is a crucial skill in mathematics and problem-solving in general. By mastering these methods, we are better equipped to tackle more complex mathematical challenges and apply these skills in real-world scenarios. The solutions, x = 8 and x = -2, represent the points where the quadratic function intersects the x-axis, providing a visual representation of the algebraic solutions. This connection between algebraic and graphical representations further deepens our understanding of quadratic equations and their applications.

Final Answer: The final answer is (A) x=-2 or x=8