Solving For X In The Equation (x-2)^2 + 1 = 2x - 3
Understanding the Problem
At the heart of this problem lies a fundamental question in algebra: solving for x. We are presented with an equation, (x-2)^2 + 1 = 2x - 3, and our mission is to find the value or values of 'x' that make this equation true. This task involves unraveling the equation, simplifying it, and employing algebraic techniques to isolate 'x' on one side. Before we dive into the step-by-step solution, it's crucial to appreciate the broader context of such problems. Solving equations is a cornerstone of mathematics, with applications spanning various fields, from physics and engineering to economics and computer science. Mastering these skills equips us with the tools to model real-world scenarios, make predictions, and solve complex problems. The equation we're tackling here is a quadratic equation in disguise, meaning it can be rearranged into the standard quadratic form of ax^2 + bx + c = 0. Recognizing this is key because we have well-established methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Each of these methods offers a different pathway to the solution, and the choice of method often depends on the specific characteristics of the equation. For instance, factoring is efficient when the quadratic expression can be easily factored, while the quadratic formula provides a universal solution, regardless of the equation's complexity. In our case, we'll first expand and simplify the equation to bring it into the standard quadratic form. This will involve applying the distributive property, combining like terms, and rearranging the equation. Once we have the equation in the standard form, we can then decide on the most appropriate method for solving it. Whether we opt for factoring, completing the square, or the quadratic formula, the goal remains the same: to find the values of 'x' that satisfy the original equation. As we proceed through the solution, we'll emphasize each step, explaining the underlying logic and the algebraic principles at play. This will not only help you understand the solution to this specific problem but also equip you with the knowledge and skills to tackle similar equations in the future. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the reasoning behind it. This approach will foster a deeper understanding of the subject and build your problem-solving abilities.
Step-by-Step Solution
Let's embark on the journey of solving for x in the equation (x-2)^2 + 1 = 2x - 3. Our first step involves expanding the squared term, (x-2)^2. Remember, this means (x-2) multiplied by itself. Applying the distributive property (also known as FOIL – First, Outer, Inner, Last), we get:
(x-2)(x-2) = xx - 2x - 2*x + 4 = x^2 - 4x + 4
Now, we substitute this back into our original equation:
x^2 - 4x + 4 + 1 = 2x - 3
Next, we simplify the left side by combining the constant terms:
x^2 - 4x + 5 = 2x - 3
To bring the equation into the standard quadratic form (ax^2 + bx + c = 0), we need to move all terms to one side. Let's subtract 2x from both sides:
x^2 - 4x - 2x + 5 = -3
Which simplifies to:
x^2 - 6x + 5 = -3
Now, add 3 to both sides:
x^2 - 6x + 5 + 3 = 0
This gives us our quadratic equation in standard form:
x^2 - 6x + 8 = 0
Now that we have the equation in standard form, we can explore different methods for solving it. Factoring is often the quickest route if the quadratic expression can be easily factored. In this case, we're looking for two numbers that multiply to 8 and add up to -6. Those numbers are -2 and -4. So, we can factor the equation as follows:
(x - 2)(x - 4) = 0
For the product of two factors to be zero, at least one of them must be zero. Therefore, we set each factor equal to zero and solve for x:
x - 2 = 0 or x - 4 = 0
Solving these simple equations, we get:
x = 2 or x = 4
Thus, the solutions to the equation (x-2)^2 + 1 = 2x - 3 are x = 2 and x = 4. We have successfully unraveled the equation and found the values of 'x' that make it true. This step-by-step approach not only provides the solution but also highlights the underlying algebraic principles and techniques involved in solving quadratic equations. Remember, the ability to solve equations is a fundamental skill in mathematics and opens doors to tackling a wide range of problems in various disciplines.
Verification of Solutions
After diligently solving for x, it's crucial to verify our solutions. This step ensures that the values we've obtained are indeed correct and satisfy the original equation. Plugging the solutions back into the equation serves as a check against potential errors made during the solving process. It also deepens our understanding of the equation and the behavior of its solutions.
Let's begin by verifying the first solution, x = 2. We substitute x = 2 into the original equation, (x-2)^2 + 1 = 2x - 3:
(2-2)^2 + 1 = 2(2) - 3
Simplifying the left side, we have:
(0)^2 + 1 = 1
Simplifying the right side, we have:
4 - 3 = 1
Since both sides are equal (1 = 1), the solution x = 2 is verified. This confirms that when x is 2, the equation holds true.
Now, let's verify the second solution, x = 4. We substitute x = 4 into the original equation:
(4-2)^2 + 1 = 2(4) - 3
Simplifying the left side, we have:
(2)^2 + 1 = 4 + 1 = 5
Simplifying the right side, we have:
8 - 3 = 5
Again, both sides are equal (5 = 5), so the solution x = 4 is also verified. This confirms that when x is 4, the equation holds true as well.
The verification process has demonstrated that both x = 2 and x = 4 are valid solutions to the equation (x-2)^2 + 1 = 2x - 3. This not only gives us confidence in our solutions but also reinforces the importance of verification in problem-solving. It's a practice that should be adopted whenever possible, especially in mathematics, where accuracy is paramount. Moreover, verification helps us catch any mistakes we might have made along the way, allowing us to correct them and learn from them. In this case, the successful verification of both solutions underscores the correctness of our approach and the validity of the solutions we've found. This thoroughness is a hallmark of good mathematical practice and contributes to a deeper understanding of the concepts involved.
Alternative Methods for Solving Quadratic Equations
While factoring proved to be an efficient method for solving for x in this particular equation, it's essential to recognize that alternative methods exist for tackling quadratic equations. These methods, such as completing the square and the quadratic formula, offer different pathways to the solution and can be particularly useful when factoring is not straightforward. Understanding these methods expands our problem-solving toolkit and allows us to choose the most appropriate approach for a given equation.
Completing the Square
Completing the square is a technique that transforms a quadratic equation into a perfect square trinomial, which can then be easily solved. This method involves manipulating the equation to create a squared term on one side and a constant on the other. Let's revisit our equation, x^2 - 6x + 8 = 0, and apply the completing the square method.
First, we move the constant term to the right side of the equation:
x^2 - 6x = -8
Next, we take half of the coefficient of the x term (-6), square it ((-3)^2 = 9), and add it to both sides of the equation:
x^2 - 6x + 9 = -8 + 9
This transforms the left side into a perfect square trinomial:
(x - 3)^2 = 1
Now, we take the square root of both sides:
x - 3 = ±1
Finally, we solve for x:
x = 3 ± 1
This gives us two solutions:
x = 3 + 1 = 4
x = 3 - 1 = 2
As we can see, completing the square yields the same solutions as factoring, x = 2 and x = 4.
The Quadratic Formula
The quadratic formula is a universal solution for quadratic equations in the standard form ax^2 + bx + c = 0. It provides a direct way to find the solutions, regardless of whether the equation can be easily factored or not. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, x^2 - 6x + 8 = 0, we have a = 1, b = -6, and c = 8. Plugging these values into the quadratic formula, we get:
x = (6 ± √((-6)^2 - 4 * 1 * 8)) / (2 * 1)
Simplifying:
x = (6 ± √(36 - 32)) / 2
x = (6 ± √4) / 2
x = (6 ± 2) / 2
This gives us two solutions:
x = (6 + 2) / 2 = 4
x = (6 - 2) / 2 = 2
Again, the quadratic formula confirms our solutions, x = 2 and x = 4. This demonstrates the power and versatility of the quadratic formula as a tool for solving quadratic equations.
Conclusion
In conclusion, the journey of solving for x in the equation (x-2)^2 + 1 = 2x - 3 has been a comprehensive exploration of algebraic techniques and problem-solving strategies. We began by expanding and simplifying the equation, transforming it into the standard quadratic form. From there, we employed factoring as our primary method, successfully finding the solutions x = 2 and x = 4. To reinforce our understanding and broaden our toolkit, we delved into alternative methods, including completing the square and the quadratic formula, both of which corroborated our initial findings.
The verification step further solidified our confidence in the solutions, demonstrating the importance of checking our work to ensure accuracy. This meticulous approach not only confirms the correctness of our answers but also deepens our grasp of the underlying mathematical principles. Throughout this process, we've emphasized the significance of understanding each step, the reasoning behind it, and the connection to broader mathematical concepts.
Solving quadratic equations is a fundamental skill in mathematics, with applications extending far beyond the classroom. It's a skill that empowers us to model real-world phenomena, make predictions, and solve complex problems across various disciplines. By mastering these techniques, we equip ourselves with a powerful tool for navigating the mathematical landscape. Moreover, the exploration of different solution methods highlights the flexibility and richness of mathematics. There's often more than one way to approach a problem, and the choice of method can depend on personal preference, the specific characteristics of the equation, or the context of the problem.
The ability to choose the most efficient and effective method is a hallmark of a skilled problem-solver. As we continue our mathematical journey, we'll encounter a myriad of problems, each with its unique challenges and opportunities. The key to success lies not only in mastering the individual techniques but also in developing a holistic understanding of the subject and a flexible, adaptable approach to problem-solving. The process of solving for x in this equation has served as a valuable exercise in this regard, reinforcing our understanding of quadratic equations and equipping us with the skills and confidence to tackle future mathematical endeavors.
