Complete The Square And Write The Given Equation In Standard Form. Then, Give The Center And Radius Of The Circle And Graph The Equation.${x^2 + Y^2 - 4x - 8y - 5 = 0}$

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Introduction to Completing the Square

Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, known as the standard form. This technique is particularly useful when dealing with equations of circles, as it allows us to easily identify the center and radius of the circle. In this article, we will learn how to complete the square and write the given equation in standard form. We will also determine the center and radius of the circle and graph the equation.

Completing the Square for the Given Equation

The given equation is x2+y2−4x−8y−5=0.{x^2 + y^2 - 4x - 8y - 5 = 0.} To complete the square, we need to group the x-terms and y-terms separately and then add and subtract the square of half the coefficient of each term.

Grouping the x-terms and y-terms

First, let's group the x-terms and y-terms separately:

x2−4x+y2−8y=5{x^2 - 4x + y^2 - 8y = 5}

Adding and Subtracting the Square of Half the Coefficient

Next, we need to add and subtract the square of half the coefficient of each term. For the x-term, the coefficient is -4, so half of it is -2. The square of -2 is 4. For the y-term, the coefficient is -8, so half of it is -4. The square of -4 is 16.

x2−4x+4+y2−8y+16=5+4+16{x^2 - 4x + 4 + y^2 - 8y + 16 = 5 + 4 + 16}

Simplifying the Equation

Now, let's simplify the equation by combining like terms:

(x−2)2+(y−4)2=25{(x - 2)^2 + (y - 4)^2 = 25}

Writing the Equation in Standard Form

The equation is now in standard form, which is (x−h)2+(y−k)2=r2,{(x - h)^2 + (y - k)^2 = r^2,} where (h, k) is the center of the circle and r is the radius.

Determining the Center and Radius of the Circle

Comparing the equation to the standard form, we can see that the center of the circle is (2, 4) and the radius is 25=5.{\sqrt{25} = 5.}

Graphing the Equation

To graph the equation, we can use the center and radius of the circle. The center of the circle is (2, 4), so we can plot this point on the coordinate plane. The radius of the circle is 5, so we can draw a circle with a radius of 5 centered at (2, 4).

Conclusion

In this article, we learned how to complete the square and write the given equation in standard form. We also determined the center and radius of the circle and graphed the equation. Completing the square is a powerful technique that can be used to solve a wide range of mathematical problems, including equations of circles.

Real-World Applications of Completing the Square

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

  • Physics: Completing the square is used to solve problems involving motion and energy.
  • Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: Completing the square is used in algorithms for solving problems involving graphs and networks.

Tips and Tricks for Completing the Square

Here are some tips and tricks for completing the square:

  • Make sure to group the x-terms and y-terms separately.
  • Add and subtract the square of half the coefficient of each term.
  • Simplify the equation by combining like terms.
  • Use the standard form to determine the center and radius of the circle.

Practice Problems

Here are some practice problems to help you master the technique of completing the square:

  • Complete the square for the equation x2+y2+2x−4y−3=0.{x^2 + y^2 + 2x - 4y - 3 = 0.}
  • Determine the center and radius of the circle for the equation (x−3)2+(y+2)2=16.{(x - 3)^2 + (y + 2)^2 = 16.}
  • Graph the equation (x+2)2+(y−1)2=9.{(x + 2)^2 + (y - 1)^2 = 9.}

Conclusion

Completing the square is a powerful technique that can be used to solve a wide range of mathematical problems, including equations of circles. By following the steps outlined in this article, you can master the technique of completing the square and apply it to real-world problems.

Introduction

Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, known as the standard form. This technique is particularly useful when dealing with equations of circles, as it allows us to easily identify the center and radius of the circle. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, known as the standard form. This technique involves adding and subtracting a constant term to create a perfect square trinomial.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to easily identify the center and radius of a circle. It also helps us to solve problems involving motion and energy in physics, design and optimize systems in engineering, and solve problems involving graphs and networks in computer science.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

  1. Group the x-terms and y-terms separately.
  2. Add and subtract the square of half the coefficient of each term.
  3. Simplify the equation by combining like terms.
  4. Use the standard form to determine the center and radius of the circle.

Q: What is the standard form of a circle equation?

A: The standard form of a circle equation is (x−h)2+(y−k)2=r2,{(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 determine the center and radius of a circle?

A: To determine the center and radius of a circle, you need to follow these steps:

  1. Rewrite the equation in standard form.
  2. Identify the values of h and k, which are the coordinates of the center of the circle.
  3. Identify the value of r, which is the radius of the circle.

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

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

  • Not grouping the x-terms and y-terms separately.
  • Not adding and subtracting the square of half the coefficient of each term.
  • Not simplifying the equation by combining like terms.
  • Not using the standard form to determine the center and radius of the circle.

Q: How do I graph a circle equation?

A: To graph a circle equation, you need to follow these steps:

  1. Determine the center and radius of the circle.
  2. Plot the center of the circle on the coordinate plane.
  3. Draw a circle with a radius equal to the radius of the circle centered at the center of the circle.

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

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

  • Physics: Completing the square is used to solve problems involving motion and energy.
  • Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Computer Science: Completing the square is used in algorithms for solving problems involving graphs and networks.

Q: How do I practice completing the square?

A: To practice completing the square, you can try the following:

  • Complete the square for a variety of equations.
  • Determine the center and radius of a circle for a variety of equations.
  • Graph a variety of circle equations.
  • Try solving problems involving motion and energy in physics, design and optimize systems in engineering, and solve problems involving graphs and networks in computer science.

Conclusion

Completing the square is a powerful technique that can be used to solve a wide range of mathematical problems, including equations of circles. By following the steps outlined in this article, you can master the technique of completing the square and apply it to real-world problems.