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 + y² − x + 2y + 1 = 0

Verified step by step guidance

1. Start by rearranging the equation to group the x and y terms together: $x^2-x+y^2+2y=-1$.
2. To complete the square for the x terms, take the coefficient of x, which is -1, divide it by 2, and square it: $(-1/2{)}^2=1/4$. Add and subtract this inside the equation.
3. Similarly, complete the square for the y terms. Take the coefficient of y, which is 2, divide it by 2, and square it: $(2/2{)}^2=1$. Add and subtract this inside the equation.
4. Rewrite the equation with the completed squares: $(x-1/2{)}^2+(y+1{)}^2=1/4+1-1$.
5. Simplify the equation to get it into the standard form of a circle: $(x-1/2{)}^2+(y+1{)}^2=r^2$, where the center is $(1/2,-1)$ and the radius is $\sqrt{r^2}$.

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 rearranging the equation and adding a constant to both sides to create a squared binomial. It is essential for converting equations into standard form, particularly for circles and parabolas.

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) is the center of the circle and r is the radius. This form allows for easy identification of the circle's center and radius, which are crucial for graphing the circle accurately.

Graphing Circles

Graphing circles involves plotting points that satisfy the circle's equation on a coordinate plane. Understanding the center and radius from the standard form is vital, as the center indicates the circle's location, while the radius determines its size. This visual representation helps in understanding the geometric properties of the circle.