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.