Rewrite each equation in the standard form for the equation of a circle, and identify its center

Rewrite each equation in the standard form for the equation...

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square trinomial. This method is essential when rewriting quadratic equations, such as those representing circles, into a form that clearly reveals geometric properties.

Standard Form of a Circle Equation

The standard form for the equation of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. Rewriting an equation into this form makes the geometric interpretation straightforward.

Identification of Center and Radius

After rewriting a circle’s equation into its standard form, the values corresponding to h, k, and r² immediately reveal the circle's center and radius. This identification process is key in analyzing and graphing circles.

Transcript

00:03 All right, here we have the equation of a circle, and we want to change it to standard form.

00:08 And to do that, we're going to need to use an algebraic process called completing the square.

00:13 And so what we want to do is figure out what to add to make a trinomial that's a perfect square trinomial for x, and similarly, what to add to make a perfect square trinomial for y.

00:24 So i'm going to put little boxes to represent where i'm going to add something.

00:28 And then, of course, if i add something to the left side of the equation, i also have to add it to the right side of the equation.

00:34 So what do we add? so to make a perfect square trinomial, remember we take this number here, negative 5, and take half of it, and then square it.

00:42 So half of it would be negative 5 halves, and when we square it we get 25 fourths.

00:48 So 25 4ths goes in that box, so then i'm going to add it to the other side as well.

00:52 And that way i have, when i factor this, i have x minus 5 halves quantity squared.

00:58 And for the other one, to make that a perfect square trinomial, we take half of 3, which is 3 halves, and we square it, and that's 9 fourths, and that's what we add, and add that to the other side of the equation as well.

01:10 And that way we get the trinomial factored into y plus three halves quantity squared.

01:16 So let's see what we have so far.

01:19 We have x minus five halves quantity squared plus y plus three halves quantity squared equals.

01:29 Now we need to add all these things together.

01:31 We have two.

01:32 We have 25 fourths...

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