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 with the given equation: \(x^{2} - 2x + y^{2} - 15 = 0\).
2. Group the \(x\) terms and the \(y\) terms together, and move the constant to the other side: \(x^{2} - 2x + y^{2} = 15\).
3. Complete the square for the \(x\) terms. Take half of the coefficient of \(x\) (which is \(-2\)), square it, and add it to both sides: half of \(-2\) is \(-1\), and \((-1)^{2} = 1\). So add \$1\( to both sides: \)x^{2} - 2x + 1 + y^{2} = 15 + 1$.
4. Rewrite the perfect square trinomial as a binomial squared: \((x - 1)^{2} + y^{2} = 16\).
5. Identify the center and radius of the circle from the standard form \((x - h)^{2} + (y - k)^{2} = r^{2}\). Here, the center is \((1, 0)\) and the radius is \(\,\sqrt{16}\).

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² = k by adding and subtracting a constant. This technique helps transform the equation into a form that reveals geometric properties, such as the center and radius of a circle.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Writing the equation in this form makes it easy to identify these key features and understand the circle's position and size.

Graphing Circles

Graphing a circle involves plotting its center (h, k) and using the radius r to mark points at a distance r in all directions. This visual representation helps in understanding the circle's location and size on the coordinate plane.