Math Problem Statement

x²+y²-4x-14=0

Solution

Let's start by analyzing and rearranging the given equation:

x2+y24x14=0x^2 + y^2 - 4x - 14 = 0

To express this equation in the standard form of a circle (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2, we need to complete the square for the xx terms and simplify the equation.

Step 1: Rearrange the equation

Move the constant term to the other side:

x2+y24x=14x^2 + y^2 - 4x = 14

Step 2: Complete the square for the xx terms

For the xx terms:

x24xx^2 - 4x

To complete the square, take half of the coefficient of xx, which is 4-4, divide by 2 to get 2-2, and then square it to get 44.

So, add and subtract 4 inside the equation:

(x24x+4)+y2=14+4(x^2 - 4x + 4) + y^2 = 14 + 4

This simplifies to:

(x2)2+y2=18(x - 2)^2 + y^2 = 18

Step 3: Interpret the equation

Now, the equation is in the standard form of a circle:

(x2)2+y2=18(x - 2)^2 + y^2 = 18

  • Center: The center of the circle is at (2,0)(2, 0).
  • Radius: The radius rr is the square root of 18, which simplifies to r=18=32r = \sqrt{18} = 3\sqrt{2}.

Summary:

  • Center: (2,0)(2, 0)
  • Radius: 323\sqrt{2}

Would you like more details or have any further questions?

Here are some related questions:

  1. How do you graph this circle?
  2. How do you complete the square for a quadratic equation?
  3. What is the area of this circle?
  4. How do you find the circumference of the circle?
  5. How does the equation change if the center of the circle is moved to (0,0)(0, 0)?
  6. What if the radius were doubled? How would the equation look?
  7. How does completing the square help in converting equations to standard forms?
  8. What are the key differences between this equation and the general equation of a circle?

Tip: Completing the square is a valuable technique for transforming quadratic equations into more useful forms, such as the standard form of a circle.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Circle Equations

Formulas

Completing the square

Theorems

Standard form of a circle

Suitable Grade Level

Grades 10-12