In Exercises 53–64, complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² - 2x + y² – 15 = 0

Verified step by step guidance

1. Start by rearranging the equation to group the x and y terms: \(x^2 - 2x + y^2 = 15\).
2. To complete the square for the x terms, take the coefficient of x, which is -2, divide it by 2 to get -1, and then square it to get 1. Add and subtract this inside the equation: \((x^2 - 2x + 1) - 1 + y^2 = 15\).
3. Rewrite the x terms as a perfect square: \((x - 1)^2 - 1 + y^2 = 15\).
4. Move the -1 to the other side of the equation: \((x - 1)^2 + y^2 = 16\).
5. Recognize the equation \((x - 1)^2 + (y - 0)^2 = 4^2\) as the standard form of a circle with center at \((1, 0)\) and radius 4.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique involves manipulating the equation to express it in the form (x-h)² + (y-k)² = r², where (h, k) is the center of the circle and r is the radius. It simplifies the process of identifying the properties of the conic section represented by the equation.

Standard Form of a Circle

The standard form of a circle's equation is given by (x-h)² + (y-k)² = r², where (h, k) represents the center of the circle and r is the radius. This form is essential for easily identifying the circle's center and radius, which are critical for graphing the circle accurately. Understanding this format allows for quick interpretation of the circle's geometric properties.

Graphing Circles

Graphing circles involves plotting points that satisfy the circle's equation on a coordinate plane. The center (h, k) is marked first, and then points are plotted at a distance r from the center in all directions. This visual representation helps in understanding the circle's size and position relative to other geometric figures, making it easier to analyze relationships in algebraic contexts.