Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y² − x + 2y + 1 = 0

Verified step by step guidance

1. Start with the given equation: \(x^{2} + y^{2} - x + 2y + 1 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^{2} - x) + (y^{2} + 2y) = -1\) (move the constant term to the right side).
3. Complete the square for the \(x\) terms: take half of the coefficient of \(x\) (which is \(-1\)), square it, and add inside the parentheses. Half of \(-1\) is \(-\frac{1}{2}\), and its square is \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). Do the same for the \(y\) terms: half of \(2\) is \(1\), and its square is \(1^2 = 1\).
4. Add these squares to both sides of the equation to keep it balanced: \((x^{2} - x + \frac{1}{4}) + (y^{2} + 2y + 1) = -1 + \frac{1}{4} + 1\).
5. Rewrite each perfect square trinomial as a squared binomial: \((x - \frac{1}{2})^{2} + (y + 1)^{2} = \text{(simplify the right side)}\). This is the standard form of a circle equation, where the center is at \(\left(\frac{1}{2}, -1\right)\) and the radius is the square root of the right side.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting appropriate constants. This technique helps transform the general form of a circle's equation into its standard form, making it easier to identify key features like the center and radius.

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Writing the equation in this form allows for straightforward identification of the circle's geometric properties and simplifies graphing.

Identifying the Center and Radius from the Equation

Once the equation is in standard form, the center of the circle is given by the coordinates (h, k), and the radius is the square root of the constant on the right side. Understanding how to extract these values is essential for graphing the circle accurately and interpreting its position and size.