Solving X² - 10x + 25 = 35 A Step-by-Step Guide
When faced with the task of solving for x in the equation x² - 10x + 25 = 35, we delve into the realm of quadratic equations. These equations, characterized by the presence of a squared variable, often require a strategic approach to find their solutions. This article provides a comprehensive guide to solving this particular equation, while also shedding light on the broader techniques applicable to quadratic equations in general. By understanding the underlying principles and mastering the steps involved, you'll be well-equipped to tackle a wide range of mathematical challenges.
Understanding the Quadratic Equation
Our main keyword is quadratic equation, and this topic is fundamental to algebra and appears in various fields, including physics, engineering, and economics. 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 ≠ 0. The equation we are tasked with solving, x² - 10x + 25 = 35, fits this description, although it requires a bit of manipulation to bring it into the standard form. Before diving into the solution, it’s crucial to recognize the structure of a quadratic equation. The x² term indicates the second degree, the -10x term is the linear term, and 25 and 35 are constants. Recognizing this structure helps us choose the appropriate method for solving the equation. The goal is to find the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. 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, depending on the specific equation. Understanding these methods and when to apply them is key to mastering quadratic equation solving.
Transforming the Equation into Standard Form
To effectively solve for x, our main keyword here is standard form, we must first transform the given equation, x² - 10x + 25 = 35, into the standard form of a quadratic equation, which is ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. In our case, we need to subtract 35 from both sides of the equation. This step is crucial because it sets the stage for applying various solution methods, such as factoring or using the quadratic formula. Subtracting 35 from both sides of x² - 10x + 25 = 35 yields x² - 10x + 25 - 35 = 0. Simplifying the left side, we combine the constant terms 25 and -35 to get x² - 10x - 10 = 0. Now, the equation is in the standard form, where a = 1, b = -10, and c = -10. This standard form allows us to easily identify the coefficients, which are necessary for applying methods like the quadratic formula. Understanding how to manipulate equations into standard form is a fundamental skill in algebra and is essential for solving various types of equations, not just quadratics. The standard form not only simplifies the solving process but also provides a clear structure for analyzing the equation's properties, such as the discriminant, which helps determine the nature of the roots.
Completing the Square: A Powerful Technique
One effective method for solving quadratic equations is completing the square, which is our main keyword for this paragraph. This technique involves transforming the equation into a perfect square trinomial, making it easier to solve for x. While factoring is a straightforward approach when applicable, completing the square provides a reliable method even when factoring is not immediately apparent. In our equation, x² - 10x - 10 = 0, we can apply the completing the square method. The first step is to focus on the x² and x terms and create a perfect square trinomial. To do this, we take half of the coefficient of the x term (which is -10), square it, and add it to both sides of the equation. Half of -10 is -5, and (-5)² is 25. So, we add 25 to both sides of the equation: x² - 10x - 10 + 25 = 25. This simplifies to x² - 10x + 25 = 35. Notice that the left side is now a perfect square trinomial, which can be factored as (x - 5)². Thus, our equation becomes (x - 5)² = 35. The process of completing the square is not just a method for solving equations; it also provides valuable insight into the structure of quadratic equations and is used in various mathematical contexts, such as deriving the quadratic formula and solving optimization problems. By understanding completing the square, you gain a deeper appreciation for the properties of quadratic expressions and their applications.
Isolating x: Finding the Solutions
Our main keyword for this section is isolating x. Now that we have the equation in the form (x - 5)² = 35, we can solve for x by isolating x. This involves taking the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots. Taking the square root of both sides gives us √((x - 5)²) = ±√35, which simplifies to x - 5 = ±√35. To isolate x, we add 5 to both sides of the equation: x = 5 ± √35. This means that there are two possible solutions for x: x = 5 + √35 and x = 5 - √35. Isolating x is a fundamental step in solving any equation, and it requires careful attention to algebraic operations and the properties of equality. In the case of quadratic equations, the presence of the square root often leads to two solutions, reflecting the quadratic nature of the equation. The solutions we found, x = 5 + √35 and x = 5 - √35, are irrational numbers because they involve the square root of a non-perfect square (35). These solutions represent the points where the parabola described by the equation x² - 10x - 10 = 0 intersects the x-axis. Understanding the process of isolating x is crucial for solving a wide variety of equations and for developing a strong foundation in algebra.
Verifying the Solutions
The final step in solving any equation, and our main keyword for this paragraph is verifying the solutions, is to verify the solutions. It is crucial to substitute the obtained values of x back into the original equation to ensure they satisfy it. This step helps to catch any errors made during the solving process and confirms the accuracy of the solutions. In our case, we found two solutions: x = 5 + √35 and x = 5 - √35. Let's verify the solutions by substituting each value back into the original equation, x² - 10x + 25 = 35. First, substitute x = 5 + √35: (5 + √35)² - 10(5 + √35) + 25. Expanding and simplifying this expression, we get: (25 + 10√35 + 35) - (50 + 10√35) + 25 = 25 + 10√35 + 35 - 50 - 10√35 + 25 = 35. This confirms that x = 5 + √35 is a valid solution. Next, substitute x = 5 - √35: (5 - √35)² - 10(5 - √35) + 25. Expanding and simplifying, we get: (25 - 10√35 + 35) - (50 - 10√35) + 25 = 25 - 10√35 + 35 - 50 + 10√35 + 25 = 35. This also confirms that x = 5 - √35 is a valid solution. By verifying the solutions, we gain confidence in our results and ensure that we have accurately solved the equation. This practice is essential for developing strong problem-solving skills in mathematics and for avoiding errors in more complex calculations.
Conclusion
In conclusion, the solutions for the equation x² - 10x + 25 = 35 are x = 5 ± √35. We arrived at these solutions by transforming the equation into standard form, completing the square, isolating x, and verifying the solutions. This step-by-step process highlights the importance of understanding the underlying principles of quadratic equations and mastering the techniques for solving them. By following these methods, you can confidently tackle a wide range of mathematical problems and develop a deeper appreciation for the power and beauty of algebra. Remember to always verify the solutions to ensure accuracy and to reinforce your understanding of the concepts involved.
Final Answer: The final answer is
