There Are Four Steps For Converting The Equation X 2 + Y 2 + 12 X + 2 Y − 1 = 0 X^2+y^2+12x+2y-1=0 X 2 + Y 2 + 12 X + 2 Y − 1 = 0 Into Standard Form By Completing The Square. Complete The Last Step.1. Group The X X X Terms Together And The Y Y Y Terms Together, And Move The Constant Term

Step 1: Group the $x$ Terms Together and the $y$ Terms Together, and Move the Constant Term

The given equation is $x^2+y^2+12x+2y-1=0$. To convert it into standard form by completing the square, we need to group the $x$ terms together and the $y$ terms together, and move the constant term to the right-hand side of the equation.

x^2 + 12x + y^2 + 2y = 1

Step 2: Complete the Square for the $x$ Terms

To complete the square for the $x$ terms, we need to add $(12/2)^2 = 36$ to both sides of the equation.

x^2 + 12x + 36 + y^2 + 2y = 1 + 36

This can be rewritten as:

(x + 6)^2 + y^2 + 2y = 37

Step 3: Complete the Square for the $y$ Terms

To complete the square for the $y$ terms, we need to add $(2/2)^2 = 1$ to both sides of the equation.

(x + 6)^2 + y^2 + 2y + 1 = 37 + 1

This can be rewritten as:

(x + 6)^2 + (y + 1)^2 = 38

Step 4: Write the Equation in Standard Form

The equation is now in standard form, which is:

(x + 6)^2 + (y + 1)^2 = 38

This is the final step in converting the equation into standard form by completing the square.

Understanding the Standard Form

The standard form of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius. In this case, the center of the circle is $(-6, -1)$ and the radius is $\sqrt{38}$.

Conclusion

Converting the equation $x^2+y^2+12x+2y-1=0$ into standard form by completing the square involves four steps. The first step is to group the $x$ terms together and the $y$ terms together, and move the constant term to the right-hand side of the equation. The second step is to complete the square for the $x$ terms by adding $(12/2)^2 = 36$ to both sides of the equation. The third step is to complete the square for the $y$ terms by adding $(2/2)^2 = 1$ to both sides of the equation. The fourth step is to write the equation in standard form, which is $(x + 6)^2 + (y + 1)^2 = 38$. This is the final step in converting the equation into standard form by completing the square.

Real-World Applications

Completing the square is a useful technique in mathematics that has many real-world applications. It is used in various fields such as physics, engineering, and economics. For example, in physics, completing the square is used to find the equation of motion of an object under the influence of a force. In engineering, completing the square is used to design and optimize systems such as electrical circuits and mechanical systems. In economics, completing the square is used to model and analyze economic systems such as supply and demand.

Tips and Tricks

Completing the square can be a challenging technique to master, but with practice and patience, it can become second nature. Here are some tips and tricks to help you complete the square:

• Make sure to group the $x$ terms together and the $y$ terms together, and move the constant term to the right-hand side of the equation.
• Complete the square for the $x$ terms by adding $(12/2)^2 = 36$ to both sides of the equation.
• Complete the square for the $y$ terms by adding $(2/2)^2 = 1$ to both sides of the equation.
• Write the equation in standard form, which is $(x + 6)^2 + (y + 1)^2 = 38$.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to convert a quadratic equation into the standard form of a circle. It involves adding a constant term to both sides of the equation to create a perfect square trinomial.

Q: Why is completing the square important?

A: Completing the square is an important technique in mathematics because it allows us to convert quadratic equations into the standard form of a circle. This makes it easier to analyze and solve problems involving circles and other conic sections.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Group the $x$ terms together and the $y$ terms together, and move the constant term to the right-hand side of the equation.
2. Complete the square for the $x$ terms by adding $(12/2)^2 = 36$ to both sides of the equation.
3. Complete the square for the $y$ terms by adding $(2/2)^2 = 1$ to both sides of the equation.
4. Write the equation in standard form, which is $(x + 6)^2 + (y + 1)^2 = 38$.

Q: What is the standard form of a circle?

A: The standard form of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.

Q: How do I find the center and radius of a circle?

A: To find the center and radius of a circle, follow these steps:

1. Identify the standard form of the circle, which is $(x - h)^2 + (y - k)^2 = r^2$.
2. The center of the circle is $(h, k)$.
3. The radius of the circle is $r$.

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

A: Completing the square has many real-world applications, including:

• Physics: Completing the square is used to find the equation of motion of an object under the influence of a force.
• Engineering: Completing the square is used to design and optimize systems such as electrical circuits and mechanical systems.
• Economics: Completing the square is used to model and analyze economic systems such as supply and demand.

Q: What are some tips and tricks for completing the square?

A: Here are some tips and tricks for completing the square:

• Make sure to group the $x$ terms together and the $y$ terms together, and move the constant term to the right-hand side of the equation.
• Complete the square for the $x$ terms by adding $(12/2)^2 = 36$ to both sides of the equation.
• Complete the square for the $y$ terms by adding $(2/2)^2 = 1$ to both sides of the equation.
• Write the equation in standard form, which is $(x + 6)^2 + (y + 1)^2 = 38$.

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

A: Here are some common mistakes to avoid when completing the square:

• Not grouping the $x$ terms together and the $y$ terms together, and moving the constant term to the right-hand side of the equation.
• Not completing the square for the $x$ terms and the $y$ terms.
• Not writing the equation in standard form.

Q: How can I practice completing the square?

A: You can practice completing the square by working through examples and exercises in a textbook or online resource. You can also try solving problems on your own and checking your answers with a calculator or online tool.

Q: What are some resources for learning more about completing the square?

A: Here are some resources for learning more about completing the square:

• Textbooks: "Algebra" by Michael Artin, "Calculus" by Michael Spivak
• Online resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Video tutorials: 3Blue1Brown, Crash Course, Math Antics

By following these tips and resources, you can master the technique of completing the square and use it to solve a wide range of mathematical problems.