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To convert the given equation of the circle \( x^2 + y^2 + 6x + 14y + 37 = 0 \) to its standard form, we need to complete the square for both the \(x\)-terms and the \(y\)-terms.

### Steps:

1. **Group the terms for \(x\) and \(y\)**:
   \[
   (x^2 + 6x) + (y^2 + 14y) = -37
   \]

2. **Complete the square for \(x\)**:
   - Take the coefficient of \(x\) (which is 6), divide it by 2, and square it: 
     \[
     \left(\frac{6}{2}\right)^2 = 9
     \]
   - Add and subtract 9 inside the equation:
     \[
     (x^2 + 6x + 9) + (y^2 + 14y) = -37 + 9
     \]

3. **Complete the square for \(y\)**:
   - Take the coefficient of \(y\) (which is 14), divide it by 2, and square it:
     \[
     \left(\frac{14}{2}\right)^2 = 49
     \]
   - Add and subtract 49 inside the equation:
     \[
     (x^2 + 6x + 9) + (y^2 + 14y + 49) = -37 + 9 + 49
     \]

4. **Simplify**:
   - Rewrite the perfect square trinomials:
     \[
     (x + 3)^2 + (y + 7)^2 = 21
     \]

### Final Standard Form:
The equation of the circle in standard form is:
\[
(x + 3)^2 + (y + 7)^2 = 21
\]
This represents a circle with center \((-3, -7)\) and radius \(\sqrt{21}\).

Would you like further details on any of the steps or have any questions?

Here are some related questions:
1. How do you calculate the radius of a circle from its standard form equation?
2. What is the significance of completing the square in geometry?
3. How do you find the general form of a circle equation from its standard form?
4. Can you always convert a quadratic equation to the form of a circle equation?
5. How does the radius affect the graph of a circle?

**Tip**: When completing the square, always ensure the quadratic term's coefficient is 1 before proceeding!

Convert the following equation of a circle to standard form: x^2 + y^2 + 6x + 14y + 37 = 0

Learn how to convert the circle equation x^2 + y^2 + 6x + 14y + 37 = 0 into standard form. This step-by-step solution involves completing the square for both x and y terms, and provides insights into the geometry of circles. Suitable for students in Grades 9-11.

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