What is the standard form of the equation of the circle x2 + y2 + 10x - 4y - 7 = 0?

To find the standard form of the equation of the circle, we need to complete the square for both the \(x\) and \(y\) terms in the given equation.

The given equation is:

\[
x^2 + y^2 + 10x - 4y - 7 = 0
\]

### Step 1: Group the \(x\) and \(y\) terms together.

\[
(x^2 + 10x) + (y^2 - 4y) = 7
\]

### Step 2: Complete the square.

For the \(x\) terms \(x^2 + 10x\):

- Take half of 10, which is 5, and square it to get 25.
- Add and subtract 25 within the equation.

\[
(x^2 + 10x + 25 - 25)
\]

For the \(y\) terms \(y^2 - 4y\):

- Take half of \(-4\), which is \(-2\), and square it to get 4.
- Add and subtract 4 within the equation.

\[
(y^2 - 4y + 4 - 4)
\]

### Step 3: Rewrite the equation with the completed squares.

\[
(x + 5)^2 - 25 + (y - 2)^2 - 4 = 7
\]

Simplify:

\[
(x + 5)^2 + (y - 2)^2 - 29 = 7
\]

### Step 4: Move the constants to the other side.

\[
(x + 5)^2 + (y - 2)^2 = 7 + 29
\]

\[
(x + 5)^2 + (y - 2)^2 = 36
\]

### Final Answer: Standard Form

The standard form of the equation of the circle is:

\[
(x + 5)^2 + (y - 2)^2 = 36
\]

This represents a circle with center \((-5, 2)\) and radius \(6\) (since \(\sqrt{36} = 6\)).

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Would you like to go over the details, or do you have any questions? Here are some related questions to explore:

1. How do you determine the center and radius of a circle from its standard equation?
2. What is the general formula for completing the square?
3. How can you derive the equation of a circle given its center and radius?
4. What other conic sections have similar equations to circles?
5. How would the equation change if the circle was translated?

**Tip:** Completing the square is a powerful technique not only for circles but also for solving quadratic equations and working with other conic sections.

Explore how to find the standard form of the equation of a circle x^2 + y^2 + 10x - 4y - 7 = 0 by completing the square for both x and y terms. This problem covers important concepts in geometry and algebra, including completing the square and circle properties. Suitable for Grades 10-12.

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