Math Problem Statement
x²+y²-4x-14=0
Solution
Let's start by analyzing and rearranging the given equation:
To express this equation in the standard form of a circle , we need to complete the square for the terms and simplify the equation.
Step 1: Rearrange the equation
Move the constant term to the other side:
Step 2: Complete the square for the terms
For the terms:
To complete the square, take half of the coefficient of , which is , divide by 2 to get , and then square it to get .
So, add and subtract 4 inside the equation:
This simplifies to:
Step 3: Interpret the equation
Now, the equation is in the standard form of a circle:
- Center: The center of the circle is at .
- Radius: The radius is the square root of 18, which simplifies to .
Summary:
- Center:
- Radius:
Would you like more details or have any further questions?
Here are some related questions:
- How do you graph this circle?
- How do you complete the square for a quadratic equation?
- What is the area of this circle?
- How do you find the circumference of the circle?
- How does the equation change if the center of the circle is moved to ?
- What if the radius were doubled? How would the equation look?
- How does completing the square help in converting equations to standard forms?
- 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
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