To convert the given equation of a circle from its general form to its standard form, follow these steps:

### Given Equation:
\[ x^2 + y^2 - 14x + 2y = -14 \]

### Step 1: Group the \(x\) terms and the \(y\) terms.
\[ (x^2 - 14x) + (y^2 + 2y) = -14 \]

### Step 2: Complete the square for both \(x\) and \(y\) terms.
- For the \(x\) terms: 
  \[ x^2 - 14x \]
  Take half of \(-14\), square it: \(\left(\frac{-14}{2}\right)^2 = 49\).
  Add and subtract 49 inside the equation:
  \[ (x^2 - 14x + 49) - 49 \]

- For the \(y\) terms:
  \[ y^2 + 2y \]
  Take half of \(2\), square it: \(\left(\frac{2}{2}\right)^2 = 1\).
  Add and subtract 1 inside the equation:
  \[ (y^2 + 2y + 1) - 1 \]

### Step 3: Rewrite the equation with the completed squares.
\[ (x^2 - 14x + 49) + (y^2 + 2y + 1) = -14 + 49 + 1 \]

Simplifying the right side:
\[ (x - 7)^2 + (y + 1)^2 = 36 \]

### Step 4: Write the equation in standard form.
The standard form of the circle equation is:
\[ (x - 7)^2 + (y + 1)^2 = 6^2 \]

### Final Answer:
The standard form of the given circle equation is:
\[ (x - 7)^2 + (y + 1)^2 = 36 \]

This represents a circle with:
- **Center** at \((7, -1)\)
- **Radius** \( r = 6 \)

Would you like to go over any specific part of this process?

### Relative Questions:
1. How do you determine the center and radius from the standard form of a circle?
2. Can you explain the process of completing the square for quadratic expressions?
3. What are the key differences between the general form and standard form of a circle?
4. How do you verify the correctness of the standard form of a circle equation?
5. Why is it important to write equations in standard form when analyzing geometric shapes?

### Tip:
When completing the square, remember that whatever you add inside the parentheses must also be added to the other side of the equation to maintain balance.

Learn how to convert the equation x² + y² - 14x + 2y = -14 from general form to standard form of a circle. This problem covers key concepts including completing the square for quadratic expressions and circle equations. Suitable for advanced high school students.

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