Isolating X² In The Equation X² + (y - 5)² = 30 A Step-by-Step Guide

In the realm of mathematics, particularly in algebra and analytic geometry, manipulating equations to isolate specific variables is a fundamental skill. This article delves into the process of isolating $x^2$ in the given equation $x^2 + (y - 5)^2 = 30$. We will meticulously walk through each step, explaining the underlying principles and providing a clear understanding of the algebraic manipulations involved. This exploration is crucial for various mathematical applications, such as solving equations, graphing functions, and analyzing geometric shapes.

Understanding the Equation

Before we begin, let's understand the equation $x^2 + (y - 5)^2 = 30$. This equation represents a circle in the Cartesian coordinate system. The general 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 infer that the circle is centered at $(0, 5)$ and has a radius of $\sqrt{30}$. However, for the purpose of this article, our primary focus is on the algebraic manipulation required to isolate $x^2$, rather than the geometric interpretation of the equation.

The equation at hand, $x^2 + (y - 5)^2 = 30$, is a classic example of a circle's equation in standard form. Understanding its components is crucial before we even attempt to isolate $x^2$. The left-hand side of the equation consists of two squared terms: $x^2$ and $(y - 5)^2$. The term $(y - 5)^2$ represents the squared vertical distance from a point $(x, y)$ on the circle to the center of the circle, which lies on the line $y = 5$. The right-hand side, 30, represents the square of the circle's radius. This constant value dictates the overall size of the circle. When tackling algebraic manipulations, it's essential to remember the order of operations (PEMDAS/BODMAS) and to apply inverse operations to isolate the desired variable. In our case, we will need to address the squared term $(y - 5)^2$ before we can fully isolate $x^2$. Recognizing the equation's structure and the relationships between its terms sets the stage for a successful isolation of $x^2$.

Step-by-Step Isolation of x²

The process of isolating $x^2$ involves a series of algebraic steps, each designed to progressively simplify the equation and bring us closer to our goal. Here's a detailed breakdown:

1. Expanding the Squared Term

The first step is to expand the squared term $(y - 5)^2$. This is achieved using the binomial expansion formula, which states that $(a - b)^2 = a^2 - 2ab + b^2$. Applying this formula to our term, we get:

$(y - 5)^2 = y^2 - 2(y)(5) + 5^2 = y^2 - 10y + 25$

Substituting this back into our original equation, we have:

$x^2 + y^2 - 10y + 25 = 30$

2. Isolating x²

Now, to isolate $x^2$, we need to move all other terms to the right side of the equation. This is done by subtracting the terms $y^2$, $-10y$, and $25$ from both sides of the equation:

$x^2 = 30 - y^2 + 10y - 25$

3. Simplifying the Equation

The final step is to simplify the equation by combining the constant terms on the right side:

$x^2 = -y^2 + 10y + 5$

This is the final expression for $x^2$ isolated from the original equation. This step-by-step approach ensures that we maintain the equality of the equation throughout the process, leading us to the correct isolated form of $x^2$. Remember, each algebraic manipulation is performed on both sides of the equation to preserve the balance and validity of the expression.

Detailed Explanation of Each Step

To truly grasp the process of isolating $x^2$, let's dissect each step in greater detail. This will not only solidify your understanding but also equip you with the skills to tackle similar algebraic manipulations in the future.

Expanding the Squared Term: A Deeper Dive

The expansion of $(y - 5)^2$ is a critical step that leverages the binomial theorem or, more simply, the understanding of how to square a binomial. The formula $(a - b)^2 = a^2 - 2ab + b^2$ is a direct result of the distributive property of multiplication over addition and subtraction. When we apply this to $(y - 5)^2$, we are essentially multiplying $(y - 5)$ by itself: $(y - 5)(y - 5)$. This multiplication unfolds as follows:

• $y * y = y^2$
• $y * -5 = -5y$
• $-5 * y = -5y$
• $-5 * -5 = 25$

Combining these terms, we get $y^2 - 5y - 5y + 25$, which simplifies to $y^2 - 10y + 25$. This expansion is crucial because it allows us to separate the $y$ terms and the constant term, paving the way for isolating $x^2$. The expanded form reveals the individual contributions of $y$ and $5$ to the overall value of the squared expression, which is essential for subsequent algebraic manipulations. Understanding the mechanics of this expansion is fundamental to manipulating more complex algebraic expressions involving binomials and polynomials.

Isolating x²: The Art of Rearrangement

Isolating $x^2$ involves strategically rearranging the terms in the equation to get $x^2$ by itself on one side. This process relies on the fundamental principle that we can perform the same operation on both sides of an equation without changing its validity. In our case, we have $x^2 + y^2 - 10y + 25 = 30$. To isolate $x^2$, we need to eliminate the terms $y^2$, $-10y$, and $25$ from the left side. This is achieved by performing the inverse operations:

• Subtract $y^2$ from both sides: This cancels out the $y^2$ term on the left, leaving us with $x^2 - 10y + 25 = 30 - y^2$.
• Add $10y$ to both sides: This cancels out the $-10y$ term on the left, resulting in $x^2 + 25 = 30 - y^2 + 10y$.
• Subtract $25$ from both sides: This cancels out the $25$ on the left, giving us $x^2 = 30 - y^2 + 10y - 25$.

Each of these operations is carefully chosen to systematically remove terms from the left side and relocate them to the right side, ultimately isolating $x^2$. This process highlights the importance of maintaining balance in an equation and using inverse operations to achieve the desired rearrangement. The ability to strategically rearrange terms is a cornerstone of algebraic problem-solving.

Simplifying the Equation: Bringing it Together

The final step in isolating $x^2$ is to simplify the equation by combining like terms. In our case, the right side of the equation, $30 - y^2 + 10y - 25$, contains two constant terms: $30$ and $-25$. Combining these terms is a straightforward arithmetic operation: $30 - 25 = 5$. This simplification reduces the equation to its most concise and understandable form: $x^2 = -y^2 + 10y + 5$. This final equation explicitly expresses $x^2$ in terms of $y$, which is often the desired outcome in many mathematical contexts. The simplified form is not only aesthetically pleasing but also facilitates further analysis and manipulation of the equation. For instance, it makes it easier to graph the relationship between $x^2$ and $y$ or to solve for specific values of $x$ given $y$, further emphasizing the importance of simplifying algebraic expressions.

Common Mistakes to Avoid

When isolating variables in equations, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and ensure accuracy in your algebraic manipulations.

1. Incorrectly Applying the Distributive Property

A frequent error occurs when expanding squared terms like $(y - 5)^2$. Students may mistakenly apply the distributive property as $(y - 5)^2 = y^2 - 5^2$, which is incorrect. The correct expansion, as we discussed earlier, is $(y - 5)^2 = y^2 - 10y + 25$. Failing to account for the middle term (-10y in this case) is a significant error that can propagate through the rest of the solution. Always remember the binomial expansion formula or meticulously multiply the binomial by itself to avoid this mistake.

2. Not Performing Operations on Both Sides

The golden rule of algebraic manipulation is that any operation performed on one side of the equation must also be performed on the other side to maintain equality. For example, when isolating $x^2$, if you subtract $y^2$ from the left side, you must also subtract $y^2$ from the right side. Neglecting to do so will disrupt the balance of the equation and lead to a false result. This principle applies to all operations, including addition, subtraction, multiplication, and division.

3. Sign Errors

Sign errors are pervasive in algebra and can easily derail a solution. When moving terms from one side of the equation to the other, remember to change their signs. For instance, if we have $x^2 + y^2 - 10y + 25 = 30$, when moving $y^2$ to the right side, it becomes $-y^2$. A simple sign mistake can alter the entire equation and lead to an incorrect expression for $x^2$. Double-check the signs of each term as you manipulate the equation to minimize this risk.

4. Incorrectly Combining Like Terms

Simplifying the equation often involves combining like terms. This is a straightforward process, but errors can occur if terms are combined incorrectly. For example, in the expression $30 - y^2 + 10y - 25$, the constants $30$ and $-25$ can be combined to give $5$. However, it's crucial not to combine terms that are not alike, such as $-y^2$ and $10y$. Make sure to combine only constant terms with constant terms and variable terms with like variable terms.

5. Skipping Steps

While it may be tempting to skip steps to save time, this can often lead to errors, especially when dealing with complex equations. Each step in the algebraic manipulation process serves a purpose, and skipping steps increases the likelihood of making a mistake. It's better to write out each step clearly and methodically to ensure accuracy. This practice also makes it easier to review your work and identify any potential errors.

By being mindful of these common mistakes and taking the time to perform each step carefully, you can significantly improve your accuracy in isolating variables and manipulating equations.

Conclusion

Isolating $x^2$ in the equation $x^2 + (y - 5)^2 = 30$ demonstrates a fundamental algebraic skill applicable across various mathematical domains. Through the detailed steps of expanding the squared term, rearranging the equation, and simplifying the result, we arrived at the isolated form: $x^2 = -y^2 + 10y + 5$. This process not only provides the solution but also reinforces critical algebraic principles, such as the binomial theorem, the importance of maintaining equality, and the strategic use of inverse operations. By understanding these principles and avoiding common mistakes, you can confidently tackle more complex algebraic problems and deepen your mathematical proficiency. The ability to manipulate equations and isolate variables is a cornerstone of mathematical problem-solving and a valuable skill in various scientific and engineering disciplines.

This exploration of isolating $x^2$ serves as a building block for more advanced mathematical concepts. The principles discussed here extend to solving systems of equations, analyzing conic sections, and even delving into calculus and differential equations. Mastering these fundamental algebraic skills is an investment in your mathematical journey, paving the way for deeper understanding and more sophisticated problem-solving capabilities. So, embrace the challenge of algebraic manipulation, practice consistently, and watch your mathematical prowess grow.