https://file.aiquickdraw.com/mathbot/file/17267144177959jd23uvj.jpg

The equation shown in the image is:

\[
2x^2 - 10x + 2y^2 - 8y = -\frac{5}{2}
\]

This represents the equation of a circle, but it needs to be rewritten in standard form to determine the center and radius. Here's the step-by-step process:

### Step 1: Group \(x\) and \(y\) terms
Group the \(x\)-terms and the \(y\)-terms together, like this:

\[
2(x^2 - 5x) + 2(y^2 - 4y) = -\frac{5}{2}
\]

### Step 2: Factor out the 2
Factor out the 2 from both groups:

\[
2[(x^2 - 5x) + (y^2 - 4y)] = -\frac{5}{2}
\]

Next, we want the equation in the form where we can complete the square inside both parentheses.

### Step 3: Complete the square
To complete the square for \(x^2 - 5x\), take half of \(-5\), square it, and add it inside the parentheses:
- Half of \(-5\) is \(-\frac{5}{2}\), and its square is \(\frac{25}{4}\).

For \(y^2 - 4y\), take half of \(-4\), square it, and add it inside the parentheses:
- Half of \(-4\) is \(-2\), and its square is \(4\).

Add \(\frac{25}{4}\) and \(4\) to both sides to balance the equation:

\[
2[(x - \frac{5}{2})^2 + (y - 2)^2] = -\frac{5}{2} + 2 \times \left( \frac{25}{4} + 4 \right)
\]

### Step 4: Simplify the equation
Now simplify the right side. First, compute the values inside the parentheses:

\[
\frac{25}{4} + 4 = \frac{25}{4} + \frac{16}{4} = \frac{41}{4}
\]

So,

\[
- \frac{5}{2} + 2 \times \frac{41}{4} = - \frac{5}{2} + \frac{41}{2} = \frac{36}{2} = 18
\]

Thus, the equation becomes:

\[
2[(x - \frac{5}{2})^2 + (y - 2)^2] = 18
\]

### Step 5: Divide by 2
Divide both sides by 2:

\[
(x - \frac{5}{2})^2 + (y - 2)^2 = 9
\]

### Step 6: Identify the center and radius
This is now in the standard form of a circle equation:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where \((h, k)\) is the center and \(r\) is the radius. From the equation:

- The center is \(\left(\frac{5}{2}, 2\right)\).
- The radius is \(r = 3\) (since \(r^2 = 9\)).

So, the center is \((2.5, 2)\) as shown in the image, and the radius should be 3.

Let me know if you'd like further details on any step!

### 5 Related Questions:
1. How do you complete the square for quadratic terms in general?
2. How do you graph a circle given its center and radius?
3. How do you determine the equation of a circle from a graph?
4. What is the importance of standard form for circles?
5. How do transformations affect the position of a circle on a coordinate plane?

### Tip:
When completing the square, remember to balance the equation by adding the same value to both sides.

Identify the center and radius to graph a circle given its equation 2x^2 - 10x + 2y^2 - 8y = -5/2.

Learn how to find the center and radius of a circle from its equation 2x^2 - 10x + 2y^2 - 8y = -5/2 by completing the square. This step-by-step guide covers key algebraic and geometric concepts for students in Grades 9-12.

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