The general form of the equation of a circle is \( x^2 + y^2 + 42x + 38y - 47 = 0 \). To express the equation in standard form, we complete the squares for both x and y terms. The center of the circle is at the point (h, k), and its radius is r units. The general form of the equation of a circle with the same radius as the above circle is \( (x-h)^2 + (y-k)^2 = r^2 \), where (h, k) is the center of the new circle. (Note: Since the question does not provide the completed standard form, the center, or the radius, these elements are left as variables in the explanation. To give a complete sentence that includes this information, further computation would be required.)

To express the given equation of a circle in standard form, we need to complete the square for both the x and y terms. The given equation is: \[ x^2 + y^2 + 42x + 38y - 47 = 0. \]

Let's complete the square for the x terms first:

1. Group the x terms together and factor out the coefficient of \( x^2 \) (which is 1 in this case): \[ x^2 + 42x + \text{(blank)} + y^2 + 38y - 47 - \text{(blank)} = 0. \]

2. To complete the square, take half of the coefficient of x (which is 42) and square it. So, \( (42/2)^2 = 21^2 = 441 \). Add and subtract this number inside the equation to balance it: \[ x^2 + 42x + 441 + y^2 + 38y - 47 - 441 = 0. \]

3. Now let's complete the square for the y terms: Take half of the coefficient of y (which is 38) and square it. So, \( (38/2)^2 = 19^2 = 361 \). Add and subtract this number inside the equation as well: \[ x^2 + 42x + 441 + y^2 + 38y + 361 - 47 - 441 - 361 = 0. \]

4. Combine like terms and rewrite the equation: \[ (x^2 + 42x + 441) + (y^2 + 38y + 361) = 47 + 441 + 361. \]

5. Factor the perfect squares: \[ (x + 21)^2 + (y + 19)^2 = 849. \]

Now the equation is in the standard form of a circle: \[ (x - h)^2 + (y - k)^2 = r^2, \] where \( h = -21 \), \( k = -19 \), and \( r^2 = 849 \). The radius \( r \) is the positive square root of \( 849 \), which can be simplified to an exact or approximate decimal value.

So, the standard form of the circle's equation with the same radius but a different center (h, k) is: \[ (x - h)^2 + (y - k)^2 = 849. \]

Extra: The process of completing the square is a useful technique in algebra for transforming equations into a form that is easier to understand and work with, particularly when dealing with conic sections like circles, parabolas, ellipses, and hyperbolas. In the context of a circle, completing the square allows us to convert a general quadratic equation into the standard form that clearly shows the circle's center and radius.

The standard form of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\) is derived by considering the distance formula between two points in the coordinate plane. If a circle is centered at point \((h, k)\) and has a radius of \(r\), then any point \((x, y)\) on the circle satisfies the equation because the distance of this point from the center is equal to the radius.

To find the center and radius of the given circle, after completing the square, we identify the values for \(h\) and \(k\) from the equation and take the square root of the constant term on the right side of the equation to determine the radius \(r\). The radius is always a positive value because it represents a distance.

For students, remember that during the process of completing the square, it's crucial to maintain the balance of the equation. Every change to one side must be matched by an equal change to the other side to keep the equality true. When adding or subtracting terms inside the parentheses, don't forget to balance the equation by performing the same operations outside the parentheses.