To convert the given general equation of a circle into its standard form, we need to complete the square for both the $x$ and $y$ terms.

Given equation:

$x^2 + y^2 + 4x - 14y + 45 = 0$

Steps to convert it to the standard form:

1. Group the $x$ terms together and the $y$ terms together: $(x^2 + 4x) + (y^2 - 14y) = -45$

2. Complete the square for both $x$ and $y$ terms:

   • For $x^2 + 4x$:
     • Take half of the coefficient of $x$, which is 4, and then square it: $\left(\frac{4}{2}\right)^2 = 4$.
     • Add and subtract this value within the equation: $(x^2 + 4x + 4 - 4)$
   • For $y^2 - 14y$:
     • Take half of the coefficient of $y$, which is -14, and then square it: $\left(\frac{-14}{2}\right)^2 = 49$.
     • Add and subtract this value within the equation: $(y^2 - 14y + 49 - 49)$

   Now, the equation becomes: $(x^2 + 4x + 4) + (y^2 - 14y + 49) - 4 - 49 = -45$

3. Rewrite the equation with perfect squares: $(x + 2)^2 + (y - 7)^2 - 53 = -45$

4. Simplify the equation by adding 53 to both sides: $(x + 2)^2 + (y - 7)^2 = 8$

Standard form of the equation:

$(x + 2)^2 + (y - 7)^2 = 8$

This is the standard form of the given circle equation, where the center of the circle is at $(-2, 7)$ and the radius is $\sqrt{8}$.

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Would you like any more details or have any questions?

Here are some related questions:

1. How do you determine the center and radius of a circle from its standard equation?
2. What is the geometric interpretation of completing the square?
3. How do you find the equation of a circle given its center and radius?
4. What changes occur if the equation was a parabola instead of a circle?
5. Can the standard form equation be used to graph the circle? How?
6. What happens if the constant on the right side of the standard form is negative?
7. How does the equation change if the circle is translated?
8. How does this method apply to ellipses and hyperbolas?

Tip: Always check the signs when completing the square to ensure accuracy in finding the standard form.