Solving For X In Y = (x - 6)^2 - 10 Expressing In X = [?] ± √[?]
In this article, we will delve into the process of solving for x in the quadratic equation y = (x - 6)^2 - 10. This type of equation is commonly encountered in mathematics, particularly in algebra and calculus. Our goal is to express x in the form x = [?] ± √[?], which involves isolating x and understanding the nature of quadratic equations. We'll break down each step, providing a clear and concise explanation to ensure a comprehensive understanding. Mastering this skill is crucial for anyone studying mathematics, as it forms the basis for more advanced concepts. Let's embark on this mathematical journey and unravel the solution together.
Understanding the Quadratic Equation
Before we dive into the solution, let's first understand the anatomy of the given quadratic equation: y = (x - 6)^2 - 10. This equation represents a parabola, a U-shaped curve, when graphed. The equation is in vertex form, which is y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. In our case, the vertex is (6, -10). The vertex form provides valuable information about the parabola's position and shape. The value of a determines whether the parabola opens upwards (if a is positive) or downwards (if a is negative). In our equation, a is 1, indicating that the parabola opens upwards. Understanding the vertex form is essential for visualizing the equation and anticipating the nature of the solutions we are about to find. Furthermore, recognizing the vertex form allows us to quickly identify key features of the parabola, such as its axis of symmetry, which is a vertical line passing through the vertex. This understanding will aid us in solving for x and interpreting the results in the context of the parabola's graph. The solutions we find will represent the x-intercepts of the parabola when y is set to 0. By grasping these fundamental concepts, we lay a solid foundation for solving the equation effectively and confidently.
Step-by-Step Solution
Let's embark on the step-by-step solution to find x in the equation y = (x - 6)^2 - 10. Our objective is to isolate x and express it in the form x = [?] ± √[?].
Step 1: Isolate the Squared Term
First, we need to isolate the squared term, which is (x - 6)^2. To do this, we add 10 to both sides of the equation:
y + 10 = (x - 6)^2
This step is crucial as it sets the stage for undoing the square. By isolating the squared term, we prepare the equation for the next operation, which is taking the square root.
Step 2: Take the Square Root of Both Sides
Next, we take the square root of both sides of the equation to eliminate the square:
√(y + 10) = ±(x - 6)
It's essential to remember that when taking the square root, we consider both the positive and negative roots, hence the ± sign. This is because both the positive and negative square roots will result in the same squared value. This step is pivotal in solving for x as it removes the square, allowing us to isolate x further. The inclusion of both positive and negative roots ensures that we capture all possible solutions for x.
Step 3: Isolate x
Now, we isolate x by adding 6 to both sides of the equation:
x = 6 ± √(y + 10)
This is the solution for x in the desired form. We have successfully expressed x in terms of y, which is our primary goal. This form of the solution allows us to easily find the values of x for any given value of y. This step is the culmination of the previous steps, where we systematically isolated x by performing inverse operations. The solution x = 6 ± √(y + 10) represents two possible values of x for each value of y, reflecting the parabolic nature of the equation.
Step 4: Expressing x in the Form x = [?] ± √[?]
Comparing our solution x = 6 ± √(y + 10) with the desired form x = [?] ± √[?], we can identify the values to fill in the brackets. In this case, the first bracket should be filled with 6, and the expression under the square root is (y + 10). If we are solving for the x-intercepts, we set y to 0, and the equation becomes:
x = 6 ± √(0 + 10)
x = 6 ± √10
Thus, the values of x are 6 + √10 and 6 - √10. This final step clarifies the specific values that x can take, providing a concrete answer to the problem. By substituting y = 0, we find the x-intercepts of the parabola, which are the points where the parabola crosses the x-axis. These values are crucial for understanding the behavior and graph of the quadratic equation.
Practical Applications and Significance
Understanding how to solve for x in equations like y = (x - 6)^2 - 10 has numerous practical applications and significant implications across various fields. Quadratic equations, in general, model a wide range of real-world phenomena, from the trajectory of a projectile to the shape of a satellite dish. In physics, for instance, these equations are used to describe the motion of objects under the influence of gravity. The solutions for x often represent critical points, such as the time it takes for an object to reach its maximum height or the distance it travels before hitting the ground.
In engineering, quadratic equations are fundamental in designing structures, circuits, and systems. They help determine optimal dimensions, calculate stress and strain, and analyze system stability. For example, engineers might use quadratic equations to model the behavior of a bridge under different loads or to optimize the performance of an electrical circuit. The ability to solve for x in these contexts is crucial for ensuring the safety, efficiency, and reliability of engineered systems.
In mathematics, solving quadratic equations is a cornerstone of algebra and calculus. It forms the basis for understanding more complex equations and functions. The solutions for x provide insights into the behavior of the quadratic function, such as its roots, vertex, and axis of symmetry. These concepts are essential for graphing quadratic functions and analyzing their properties. Moreover, the techniques used to solve quadratic equations, such as completing the square and using the quadratic formula, are applicable to a broader range of mathematical problems.
Beyond these specific fields, the ability to solve quadratic equations cultivates critical thinking and problem-solving skills that are valuable in various aspects of life. The systematic approach required to isolate x and find its values reinforces logical reasoning and analytical abilities. These skills are transferable to other disciplines and contribute to overall intellectual development.
In conclusion, mastering the solution for x in quadratic equations like y = (x - 6)^2 - 10 is not merely an academic exercise; it is a fundamental skill with far-reaching applications and significance in science, engineering, mathematics, and beyond. The ability to solve these equations empowers individuals to understand, analyze, and solve real-world problems effectively.
Conclusion
In this comprehensive guide, we have successfully navigated the process of solving for x in the equation y = (x - 6)^2 - 10. We meticulously broke down each step, starting from isolating the squared term to ultimately expressing x in the desired form: x = 6 ± √10 (when y=0). This involved understanding the structure of the quadratic equation, applying algebraic manipulations, and interpreting the results in the context of the parabola's properties.
The ability to solve for variables in equations like this is a fundamental skill in mathematics, serving as a building block for more advanced concepts. It not only enhances problem-solving abilities but also provides a deeper understanding of mathematical relationships and their applications in real-world scenarios. By mastering these techniques, students and practitioners alike can approach complex problems with confidence and precision.
Furthermore, the process of solving quadratic equations reinforces critical thinking and analytical skills that are valuable across various disciplines. The systematic approach required to isolate x and find its values cultivates logical reasoning and attention to detail. These skills are transferable to other areas of study and contribute to overall intellectual growth.
In conclusion, the solution x = 6 ± √10 (when y=0) represents the values of x that satisfy the equation y = (x - 6)^2 - 10. This exercise underscores the importance of understanding quadratic equations and their solutions, as they are essential tools in mathematics, science, engineering, and beyond. The skills acquired through this process will undoubtedly prove beneficial in future mathematical endeavors and problem-solving challenges.
