Mrs. Culland Is Finding The Center Of A Circle Whose Equation Is X 2 + Y 2 + 6 X + 4 Y − 3 = 0 X^2+y^2+6x+4y-3=0 X 2 + Y 2 + 6 X + 4 Y − 3 = 0 By Completing The Square. Her Work Is Shown Below:$[ \begin{array}{l} x 2+y 2+6x+4y-3=0 \ x 2+6x+y 2+4y-3=0 \ (x 2+6x)+(y 2+4y)=3

Introduction

In mathematics, a circle is a set of points that are equidistant from a central point known as the center. The equation of a circle can be written in the form $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. In this article, we will show how to find the center of a circle by completing the square using the equation $x^2+y^2+6x+4y-3=0$.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This technique is useful in solving quadratic equations and finding the center of a circle. To complete the square, we need to add and subtract a constant term to make the expression a perfect square trinomial.

Let's start with the given equation $x^2+y^2+6x+4y-3=0$. We can rewrite it as:

$$x^2+6x+y^2+4y-3=0$$

Now, let's focus on the $x$ terms. We can add and subtract $(6/2)^2 = 9$ to make the expression a perfect square trinomial:

$$(x^2+6x+9)+(y^2+4y)-3-9=0$$

Simplifying the expression, we get:

$$(x+3)^2+(y^2+4y)-12=0$$

Now, let's focus on the $y$ terms. We can add and subtract $(4/2)^2 = 4$ to make the expression a perfect square trinomial:

$$(x+3)^2+(y^2+4y+4)-12-4=0$$

Simplifying the expression, we get:

$$(x+3)^2+(y+2)^2-16=0$$

Finding the Center of the Circle

Now that we have completed the square, we can easily find the center of the circle. The center of the circle is the point $(h,k)$ that makes the expression $(x-h)^2 + (y-k)^2$ a perfect square trinomial. In this case, we can see that the center of the circle is the point $(-3,-2)$.

Discussion

Mrs. Culland's work is shown below:

$$\[ \begin{array}{l} x^2+y^2+6x+4y-3=0 \\ x^2+6x+y^2+4y-3=0 \\ (x^2+6x)+(y^2+4y)=3 \end{array}$$

As we can see, Mrs. Culland has correctly completed the square to find the center of the circle. However, she has made a mistake in the last step. The correct equation should be $(x+3)^2+(y+2)^2-16=0$, not $(x^2+6x)+(y^2+4y)=3$.

Conclusion

In this article, we have shown how to find the center of a circle by completing the square using the equation $x^2+y^2+6x+4y-3=0$. We have also discussed Mrs. Culland's work and pointed out the mistake in the last step. By completing the square, we can easily find the center of a circle and write the equation of the circle in the form $(x-h)^2 + (y-k)^2 = r^2$.

Example Problems

1. Find the center of the circle whose equation is $x^2+y^2-4x-2y+4=0$.
2. Find the center of the circle whose equation is $x^2+y^2+2x-6y-7=0$.

Answer Key

1. The center of the circle is the point $(2,1)$.
2. The center of the circle is the point $(-1,3)$.

Glossary

• Completing the square: A technique used to rewrite a quadratic expression in the form of a perfect square trinomial.
• Perfect square trinomial: A quadratic expression that can be written in the form $(x-h)^2 + (y-k)^2$.
• Center of a circle: The point $(h,k)$ that makes the expression $(x-h)^2 + (y-k)^2$ a perfect square trinomial.
  Q&A: Finding the Center of a Circle by Completing the Square ===========================================================

Introduction

In our previous article, we showed how to find the center of a circle by completing the square using the equation $x^2+y^2+6x+4y-3=0$. In this article, we will answer some frequently asked questions about finding the center of a circle by completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This technique is useful in solving quadratic equations and finding the center of a circle.

Q: How do I complete the square?

A: To complete the square, you need to add and subtract a constant term to make the expression a perfect square trinomial. The constant term is half of the coefficient of the linear term squared.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be written in the form $(x-h)^2 + (y-k)^2$.

Q: How do I find the center of a circle by completing the square?

A: To find the center of a circle by completing the square, you need to rewrite the equation of the circle in the form $(x-h)^2 + (y-k)^2 = r^2$. The center of the circle is the point $(h,k)$.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not adding and subtracting the constant term correctly
• Not squaring the coefficient of the linear term correctly
• Not rewriting the equation in the correct form

Q: Can I use completing the square to solve quadratic equations?

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the quadratic equation in the form $(x-h)^2 + (y-k)^2 = 0$, you can easily find the solutions to the equation.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

• Finding the center of a circle in geometry
• Solving quadratic equations in algebra
• Modeling real-world problems in physics and engineering

Q: How do I practice completing the square?

A: You can practice completing the square by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some tips for mastering completing the square?

A: Some tips for mastering completing the square include:

• Practice regularly to build your skills and confidence
• Use online resources and practice tests to help you improve your skills
• Review and practice completing the square in different contexts, such as geometry and algebra

Conclusion

In this article, we have answered some frequently asked questions about finding the center of a circle by completing the square. We hope that this article has been helpful in clarifying any confusion and providing additional guidance on this important topic.

Example Problems

1. Find the center of the circle whose equation is $x^2+y^2-4x-2y+4=0$.
2. Find the center of the circle whose equation is $x^2+y^2+2x-6y-7=0$.

Answer Key

1. The center of the circle is the point $(2,1)$.
2. The center of the circle is the point $(-1,3)$.

Glossary

• Completing the square: A technique used to rewrite a quadratic expression in the form of a perfect square trinomial.
• Perfect square trinomial: A quadratic expression that can be written in the form $(x-h)^2 + (y-k)^2$.
• Center of a circle: The point $(h,k)$ that makes the expression $(x-h)^2 + (y-k)^2$ a perfect square trinomial.