Graphing The Equation $x^2 - 6x + Y^2 + 2y + 6 = 0$ A Comprehensive Guide

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In this article, we will delve into the equation $x^2 - 6x + y^2 + 2y + 6 = 0$ to determine the nature of its graph. Our primary goal is to identify the geometric shape represented by this equation. To achieve this, we'll employ the technique of completing the square, a powerful algebraic method that allows us to rewrite the equation in a more recognizable form. This process will unveil the equation's underlying structure, enabling us to pinpoint the graph's characteristics, such as its center and radius (if it's a circle), or other defining features if it represents a different conic section.

Understanding the Equation

At first glance, the equation $x^2 - 6x + y^2 + 2y + 6 = 0$ might seem complex. However, by applying algebraic manipulations, we can transform it into a standard form that reveals its geometric essence. The presence of both $x^2$ and $y^2$ terms, along with linear terms in $x$ and $y$, suggests that the equation might represent a circle or another conic section. To confirm this, we'll employ the method of completing the square.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a more convenient form. For an expression of the form $x^2 + bx$, we add and subtract $(b/2)^2$ to create a perfect square trinomial. This allows us to factor the expression into the form $(x + b/2)^2$ plus a constant term. We'll apply this method to both the $x$ and $y$ terms in our equation.

Completing the Square for x Terms

Consider the $x$ terms: $x^2 - 6x$. To complete the square, we take half of the coefficient of the $x$ term (-6), which is -3, and square it: $(-3)^2 = 9$. We then add and subtract 9 from the expression:

$x^2 - 6x = x^2 - 6x + 9 - 9 = (x - 3)^2 - 9$

Completing the Square for y Terms

Now, let's focus on the $y$ terms: $y^2 + 2y$. Half of the coefficient of the $y$ term (2) is 1, and squaring it gives us $1^2 = 1$. Adding and subtracting 1, we get:

$y^2 + 2y = y^2 + 2y + 1 - 1 = (y + 1)^2 - 1$

Rewriting the Equation

Substituting the completed square forms back into the original equation, we have:

$(x - 3)^2 - 9 + (y + 1)^2 - 1 + 6 = 0$

Simplifying, we get:

$(x - 3)^2 + (y + 1)^2 - 4 = 0$

Adding 4 to both sides, we obtain:

$(x - 3)^2 + (y + 1)^2 = 4$

Identifying the Graph

The equation $(x - 3)^2 + (y + 1)^2 = 4$ is now in the standard form of the equation of a circle, which is:

$(x - h)^2 + (y - k)^2 = r^2$

where (h, k) represents the center of the circle and r is the radius.

Center and Radius

Comparing our equation $(x - 3)^2 + (y + 1)^2 = 4$ to the standard form, we can identify the center and radius of the circle:

• Center: (h, k) = (3, -1)
• Radius: $r^2 = 4$, so $r = \sqrt{4} = 2$

Conclusion

Therefore, the equation $x^2 - 6x + y^2 + 2y + 6 = 0$ represents a circle with center (3, -1) and radius 2. This completes our analysis of the equation and its graphical representation.

Visualizing the Circle

To further solidify our understanding, let's visualize the circle. Imagine a coordinate plane. The center of the circle is located at the point (3, -1). The radius of the circle is 2 units, meaning that every point on the circle is 2 units away from the center. If you were to plot all such points, you would see a circle forming around the center.

Plotting Key Points

To get a better sense of the circle's shape, we can plot a few key points. Start with the center (3, -1). Then, move 2 units in each direction: up, down, left, and right.

• Moving 2 units to the right from the center, we reach the point (5, -1).
• Moving 2 units to the left, we reach the point (1, -1).
• Moving 2 units up, we reach the point (3, 1).
• Moving 2 units down, we reach the point (3, -3).

These four points lie on the circle and help us visualize its shape.

Sketching the Circle

With the center and a few points plotted, we can sketch the circle. The circle will be centered at (3, -1) and pass through the points (5, -1), (1, -1), (3, 1), and (3, -3). It's a circle with a radius of 2 units.

Alternative Methods

While completing the square is a direct and effective method for identifying the graph of the equation, there are alternative approaches we could consider. These methods might not be as straightforward but can provide additional insights into the equation's properties.

Using the General Equation of a Conic Section

The general equation of a conic section is given by:

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

By comparing this general form with our equation $x^2 - 6x + y^2 + 2y + 6 = 0$, we can identify the coefficients:

• A = 1
• B = 0
• C = 1
• D = -6
• E = 2
• F = 6

Since A = C and B = 0, this indicates that the equation represents a circle. This confirms our earlier conclusion based on completing the square.

Analyzing the Discriminant

Another way to classify conic sections is by analyzing the discriminant, which is given by $B^2 - 4AC$. In our case, B = 0, A = 1, and C = 1, so the discriminant is:

$0^2 - 4(1)(1) = -4$

A negative discriminant indicates that the conic section is either an ellipse or a circle. Since A = C, we can further conclude that it is a circle. This provides additional support for our findings.

Common Mistakes to Avoid

When working with equations of circles and completing the square, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and solve problems more accurately.

Incorrectly Completing the Square

The most common mistake is making an error in the process of completing the square. This can involve miscalculating the value to add and subtract, or incorrectly factoring the perfect square trinomial. For example, when completing the square for $x^2 - 6x$, a mistake might be to add and subtract the wrong value, such as 6 instead of 9. Always remember to take half of the coefficient of the linear term and square it.

Forgetting to Adjust the Constant Term

Another common error is forgetting to adjust the constant term when moving the added and subtracted values to the other side of the equation. For instance, after completing the square, we had $(x - 3)^2 - 9 + (y + 1)^2 - 1 + 6 = 0$. It's crucial to combine the constant terms correctly to obtain the correct radius of the circle. A mistake here can lead to an incorrect determination of the radius.

Misidentifying the Center and Radius

Once the equation is in standard form, $(x - h)^2 + (y - k)^2 = r^2$, it's essential to correctly identify the center (h, k) and the radius r. Remember that the coordinates of the center are the opposite of the values inside the parentheses. For example, in the equation $(x - 3)^2 + (y + 1)^2 = 4$, the center is (3, -1), not (-3, 1). Also, remember that r is the square root of the constant term on the right side of the equation.

Algebraic Errors

Simple algebraic errors, such as incorrect distribution or combining like terms improperly, can also lead to mistakes. It's always a good practice to double-check each step of your calculations to ensure accuracy.

Real-World Applications of Circles

The concept of circles extends far beyond the realm of mathematics and finds numerous applications in various real-world scenarios. Understanding circles is crucial in fields like engineering, physics, architecture, and even everyday life.

Engineering and Design

In engineering, circles are fundamental in the design of various mechanical components, such as gears, wheels, and axles. The circular shape ensures smooth rotation and efficient transmission of power. Engineers need to calculate the dimensions of these components accurately, using concepts like circumference, area, and the equation of a circle.

Physics

In physics, circular motion is a fundamental concept. Objects moving in circular paths, like planets orbiting the sun or a ball swung in a circle, are governed by principles that involve circles. Understanding the radius and center of the circular path is crucial for calculating velocity, acceleration, and other physical quantities.

Architecture

Architects often incorporate circular designs into buildings and structures for aesthetic and structural reasons. Domes, arches, and circular windows are common architectural features. The geometry of circles is essential for ensuring the stability and visual appeal of these designs.

Everyday Life

In everyday life, we encounter circles in countless ways. From the wheels of our vehicles to the circular faces of clocks and watches, circles are ubiquitous. Even in sports, such as basketball (the hoop) and track and field (the circular tracks), circles play a significant role.

Conclusion

In summary, we've thoroughly examined the equation $x^2 - 6x + y^2 + 2y + 6 = 0$ and determined that it represents a circle with a center at (3, -1) and a radius of 2. We achieved this by employing the method of completing the square, which allowed us to rewrite the equation in the standard form of a circle's equation. Additionally, we explored alternative methods, such as using the general equation of a conic section and analyzing the discriminant, to confirm our findings.

We also discussed common mistakes to avoid when working with circles and completing the square, emphasizing the importance of careful calculations and accurate identification of the center and radius. Finally, we highlighted the numerous real-world applications of circles, underscoring their significance in various fields and everyday life. Understanding the geometry of circles is not only a fundamental mathematical skill but also a valuable tool for solving practical problems and appreciating the world around us.