Isolating Y Squared In (x+4)^2 + Y^2 = 22 Solution And Explanation

Introduction

In this article, we will delve into the process of isolating $y^2$ in the given equation $(x+4)^2 + y^2 = 22$. This is a fundamental algebraic manipulation skill that is crucial for solving various mathematical problems, particularly those involving circles and other conic sections. Isolating a variable, like $y^2$ in this case, means rearranging the equation so that the desired variable is alone on one side of the equals sign. This allows us to express the variable in terms of other variables and constants, providing valuable insights into the relationship between them. By the end of this guide, you will have a clear understanding of the steps involved in isolating $y^2$, as well as the underlying mathematical principles.

Understanding the Equation

Before we begin, let's take a closer look at the equation $(x+4)^2 + y^2 = 22$. This equation represents a circle in the Cartesian coordinate system. The general form of the equation of a circle with center $(h, k)$ and radius $r$ is given by $(x-h)^2 + (y-k)^2 = r^2$. Comparing this with our given equation, we can see that the center of the circle is $(-4, 0)$ and the radius squared, $r^2$, is 22. Thus, the radius $r$ is $\sqrt{22}$. Understanding the geometrical interpretation of the equation can often provide a better intuition for the algebraic manipulations we perform. When we isolate $y^2$, we are essentially rewriting the equation in a form that highlights the relationship between the $y$-coordinate and the $x$-coordinate of points on the circle. This can be particularly useful when we want to analyze the vertical behavior of the circle or when we want to solve for $y$ in terms of $x$. Therefore, having a solid grasp of both the algebraic and geometric aspects of the equation is essential for mastering the process of isolating variables.

Step-by-Step Solution

Now, let's go through the step-by-step solution to isolate $y^2$ in the equation $(x+4)^2 + y^2 = 22$. This process involves algebraic manipulation to get $y^2$ by itself on one side of the equation.

Step 1: Expand the squared term

The first step is to expand the term $(x+4)^2$. Using the binomial expansion formula or the FOIL method (First, Outer, Inner, Last), we have:

$(x+4)^2 = (x+4)(x+4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16$

So, the equation becomes:

$x^2 + 8x + 16 + y^2 = 22$

Step 2: Isolate $y^2$

To isolate $y^2$, we need to subtract the terms $x^2$, $8x$, and $16$ from both sides of the equation. This will leave $y^2$ alone on the left side:

$y^2 = 22 - (x^2 + 8x + 16)$

Step 3: Simplify the equation

Now, we simplify the right side of the equation by distributing the negative sign and combining like terms:

$y^2 = 22 - x^2 - 8x - 16$

Combining the constant terms $22$ and $-16$, we get:

$y^2 = -x^2 - 8x + 6$

Therefore, the result of isolating $y^2$ in the equation $(x+4)^2 + y^2 = 22$ is $y^2 = -x^2 - 8x + 6$. This final equation expresses $y^2$ explicitly in terms of $x$, which is a crucial step in further analysis or problem-solving involving this equation.

Analyzing the Options

Now that we have derived the expression for $y^2$, let's compare it with the given options to identify the correct answer. The options provided are:

A. $y^2 = -x^2 - 8x + 16$ B. $y^2 = x^2 - 8x + 6$ C. $y^2 = -x^2 - 8x + 6$ D. $y^2 = 22 - x^2$

Comparing our derived result, $y^2 = -x^2 - 8x + 6$, with the options, we can clearly see that option C matches our solution. Options A, B, and D have different coefficients and constant terms, making them incorrect. Option A has a constant term of 16 instead of 6, while Option B has the wrong sign for the $x^2$ term. Option D is incomplete as it does not account for the expansion of $(x+4)^2$ and the subsequent subtraction of all terms from 22. Therefore, only option C accurately represents the result of isolating $y^2$ in the given equation.

Correct Answer: C. $y^2 = -x^2 - 8x + 6$

Common Mistakes to Avoid

When isolating variables in equations, several common mistakes can occur. Recognizing and avoiding these pitfalls is crucial for accurate problem-solving. One frequent error is incorrect expansion of squared terms. For example, students might mistakenly expand $(x+4)^2$ as $x^2 + 16$ without including the middle term $8x$. This can lead to an entirely different equation and an incorrect solution. Another common mistake is mishandling the negative sign when moving terms from one side of the equation to the other. For instance, failing to distribute the negative sign correctly when subtracting a group of terms can result in errors in both the coefficients and the constant term. Additionally, overlooking basic algebraic rules, such as combining only like terms, can lead to incorrect simplifications. To avoid these mistakes, it is essential to practice expanding squared terms carefully, pay close attention to signs, and double-check each step of the algebraic manipulation. Regularly reviewing and practicing these fundamental algebraic techniques can greatly improve accuracy and confidence in solving equations.

Practical Applications

Isolating variables is not just a theoretical exercise; it has numerous practical applications in various fields. In mathematics, this skill is fundamental to solving equations, graphing functions, and analyzing relationships between variables. For instance, in physics, isolating variables is essential for solving kinematic equations, calculating forces, and determining electrical circuit parameters. In engineering, it is used in designing structures, analyzing fluid dynamics, and optimizing control systems. Computer science also relies heavily on isolating variables in algorithms and data analysis. Moreover, in everyday life, isolating variables helps in making informed decisions, such as calculating loan payments, determining the best deals, and planning budgets. Mastering the technique of isolating variables enhances problem-solving skills and provides a strong foundation for advanced studies in mathematics, science, and engineering. By understanding how to manipulate equations effectively, individuals can tackle a wide range of real-world problems and gain a deeper appreciation for the power of algebra.

Conclusion

In conclusion, isolating $y^2$ in the equation $(x+4)^2 + y^2 = 22$ is a fundamental algebraic manipulation that demonstrates key principles of equation solving. By expanding the squared term, moving terms across the equals sign, and simplifying the result, we arrived at the solution $y^2 = -x^2 - 8x + 6$. This process underscores the importance of paying attention to detail and following algebraic rules carefully. The correct answer among the given options was C. This exercise not only reinforces algebraic skills but also highlights the practical applications of these skills in various fields. Mastering the ability to isolate variables is crucial for solving more complex problems and understanding the relationships between different quantities. Therefore, continuous practice and a solid grasp of basic algebraic principles are essential for success in mathematics and related disciplines. By understanding the underlying concepts and practicing regularly, one can develop confidence and proficiency in algebraic manipulations.